ABSTRACT
Title of dissertation: GENERALIZATIONS OF
SCHOTTKY GROUPS
Jean-Philippe Burelle,
Doctor of Philosophy, 2017
Dissertation directed by: Professor William M. Goldman
Department of Mathematics
Schottky groups are classical examples of free groups acting properly discon-
tinuously on the complex projective line. We discuss two different applications of
similar constructions. The first gives examples of 3-dimensional Lorentzian Kleinian
groups which act properly discontinuously on an open dense subset of the Einstein
universe. The second gives a large class of examples of free subgroups of automor-
phisms groups of partially cyclically ordered spaces. We show that for a certain
cyclic order on the Shilov boundary of a Hermitian symmetric space, this construc-
tion corresponds exactly to representations of fundamental groups of surfaces with
boundary which have maximal Toledo invariant.
GENERALIZATIONS OF SCHOTTKY GROUPS
by
Jean-Philippe Burelle
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2017
Advisory Committee:
Professor William M. Goldman, Chair/Advisor
Professor Scott A. Wolpert
Professor Christian Zickert
Professor John J. Millson
Professor Theodore A. Jacobson
c© Copyright by
Jean-Philippe Burelle
2017
Acknowledgments
I gratefully acknowledge support from the Natural Sciences and Engineering
Research Council of Canada (NSERC). The financial support provided by the post-
graduate fellowship made it possible for me to focus solely on research and produce
this thesis in a much shorter time than would have been possible otherwise. Addi-
tionally, I am thankful for the support of the National Science Foundation (NSF)
through the GEAR network. GEAR allowed me to meet and work with dozens of
mathematicians with similar research interests from all over the world.
I would like to thank my advisor Bill Goldman for all the opportunities that
he provided me with. Bill has an amazing ability to create networks of mathemat-
ical collaborations, and in particular, he has introduced me to almost every other
mathematician mentioned in this section.
I am thankful to every collaborator and colleague who has helped me explore
the mathematical landscape over the past five years : Dominik Francoeur, Nicolaus
Treib, Virginie Charette, Todd Drumm, Sean Lawton, Ryan Kirk, Jeffrey Danciger
and many more. I also want to thank the officemates who made graduate school
an amazing experience : Dan Zollers, Nathan Dykas, Danul Gunatilleka, Robert
Maschal, Nathaniel Monson and Mark Magsino.
Finally I would like to thank my family : my parents Andre´ and Guylaine, my
sister Cynthia and my significant other Catherine who have all been supportive and
encouraging throughout the duration of my studies. Thank you all for putting up
with me for all this time.
ii
Table of Contents
List of Figures v
1 Introduction 1
1.1 Fuchsian Schottky groups . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Lorentzian geometries 7
2.1 Models for 3-dimensional Lorentzian space forms . . . . . . . . . . . . 7
2.1.1 Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.2 Anti de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.3 de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 The Einstein universe . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 The projective model . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Geometric objects . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 The Lagrangian Grassmannian model . . . . . . . . . . . . . . 16
2.2.4 The Lie circles model . . . . . . . . . . . . . . . . . . . . . . . 22
3 Crooked Schottky groups 27
3.1 Einstein tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 Pairs of positive vectors . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Involutions in Einstein tori . . . . . . . . . . . . . . . . . . . . 29
3.2 The Lagrangian Grassmannian model . . . . . . . . . . . . . . . . . . 32
3.2.1 Nondegenerate planes and symplectic splittings . . . . . . . . 35
3.2.2 Graphs of linear maps . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Disjoint crooked surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Anti de Sitter crooked planes . . . . . . . . . . . . . . . . . . . . . . 46
3.4.1 AdS as a subspace of Ein . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Crooked surfaces and AdS crooked planes . . . . . . . . . . . 48
3.4.2.1 AdS crooked planes based at the identity . . . . . . . 48
3.4.2.2 AdS crooked planes based at f . . . . . . . . . . . . 49
3.4.3 Disjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
iii
4 Partial cyclic orders 54
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Generalized Schottky groups . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.1 Definition of generalized Schottky group . . . . . . . . . . . . 58
4.2.2 Limit curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Hermitian symmetric spaces of tube type . . . . . . . . . . . . . . . . 66
4.3.1 The partial cyclic order on the Shilov boundary . . . . . . . . 67
4.3.2 Maximal representations . . . . . . . . . . . . . . . . . . . . . 74
4.4 Schottky groups in Sp(2n,R) . . . . . . . . . . . . . . . . . . . . . . 76
4.4.1 The Maslov index in Sp(2n,R) . . . . . . . . . . . . . . . . . 76
4.4.2 Fundamental domains . . . . . . . . . . . . . . . . . . . . . . 79
4.4.2.1 Positive halfspaces and fundamental domains . . . . 80
4.4.2.2 Intervals as symmetric spaces . . . . . . . . . . . . . 85
4.4.2.3 The Riemannian distance on intervals . . . . . . . . 87
4.4.2.4 The domain of discontinuity . . . . . . . . . . . . . . 88
4.5 Oriented flags in three dimensions . . . . . . . . . . . . . . . . . . . . 92
4.5.1 Hyperconvex configurations . . . . . . . . . . . . . . . . . . . 92
Bibliography 98
iv
List of Figures
2.1 A photon in the circles model of the Einstein universe. . . . . . . . . 25
2.2 A spacelike circle in the circles model. The orientations on the outer
and inner circle are opposite, and the orientations of all the other
circles match that of the outer circle. . . . . . . . . . . . . . . . . . . 26
3.1 The three possible types of intersection for a pair of Einstein tori,
viewed in a Minkowski patch. . . . . . . . . . . . . . . . . . . . . . . 30
4.1 A combinatorial model for the once punctured torus. . . . . . . . . . 59
4.2 Some first, second and third order intervals for a generalized Schottky
group acting on S1 × S1. . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Four intervals between Lagrangians in increasing order. . . . . . . . . 78
4.4 A pair of disjoint positive halfspaces in RP3 . . . . . . . . . . . . . . 83
4.5 The first two generations of positive halfspaces for a two-generator
Schottky group in Sp(4,R). . . . . . . . . . . . . . . . . . . . . . . . 84
4.6 A hyperconvex configuration of three oriented flags, projected to the
2-sphere of directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
v
Chapter 1: Introduction
This thesis is focused around the idea of Schottky groups and different ways to
generalize their classical construction. Such generalizations provide large classes of
examples of geometrically interesting free, discrete subgroups of Lie groups. Perhaps
the most commonly known version of this arises in the theory of Kleinian groups.
A classical Schottky group is a discrete subgroup Γ ⊂ PSL(2,C) of Mo¨bius trans-
formations with an explicit presentation as follows. Choose D±1 , . . . , D
±
k disjoint,
closed round disks in CP1 and elements g1, . . . , gk ∈ PSL(2,C) such that gi maps
the interior of D−i to the complement of D
+
i . The group Γ generated by the gi is
then called a classical Schottky group. It is a free group on those k generators, and
it acts properly discontinuously on the complement of a Cantor set in CP1.
The Ping-pong lemma is the elementary result which proves that Schottky
groups are free (Lemma 1.1.2). It was introduced in its modern generality by Tits
[Tit72] in order to prove the Tits alternative. Generalizations of the construction
come in two flavors. The first is more dynamical : assume that some generators in
a linear group are sufficiently transverse and sufficiently attractive/repulsive, and
conclude that the group they generate is free. The second is more geometric, like
the construction in CP1. Natural hypersurfaces bounding disjoint halfspaces in a
1
homogeneous space are mapped pairwise by automorphisms, and these hypersurfaces
can be shown to bound a fundamental domain for the action of the resulting group.
The constructions discussed here generally belong to this second type.
Previous constructions of Schottky groups using hypersurfaces include pro-
jective linear groups acting on complex projective spaces ([Nor86], [SV03]), affine
Lorentzian groups acting on R2,1 ([Dru92]), and conformal Lorentzian groups acting
on the Einstein universe ([CFLD14], [BCFG17]).
This last example is the one we focus on in Chapter 3. The content of
this chapter is joint work with Charette, Francoeur and Goldman. In [CFLD14],
Charette, Francoeur and Lareau-Dussault explain how to construct examples of
Schottky subgroups of O(3, 2) by choosing disjoint crooked surfaces in the Ein-
stein universe. This was inspired by the success of crooked planes in the study
of Schottky groups of affine Lorentzian transformations acting on Minkowski space
([Dru92], [BCDG14], [CDGM03]). The relevance of halfspaces in this theory, as
opposed to just hypersurfaces, is motivated by the results of [BCDG14]. Namely, it
is more natural to characterize disjoint halfspaces than disjoint bounding surfaces.
Crooked surfaces were introduced by Frances [Fra03] in 2003 in order to conformally
compactify flat affine Lorentzian manifolds. The resulting subgroups of O(3, 2) are
Lorentzian Kleinian groups in the sense of Frances [Fra05]. We expand upon the
work of Charette-Francoeur-Lareau-Dussault by proving a complete disjointness cri-
terion for these surfaces, a first step towards a classification of the Schottky groups
that can be constructed in this way.
2
Theorem 1.0.1. Two crooked surfaces C1, C2 ⊂ Ein are disjoint if and only if the
four photons bounding the stem of C1 are disjoint from C2, and the four photons
bounding the stem of C2 are disjoint from C1.
This also builds upon the work of Danciger, Gue´ritaud and Kassel, who studied
crooked surfaces in the context of the negatively curved anti de Sitter space. In
particular, Theorem 1.0.1 specializes to the disjointness criterion for anti de Sitter
crooked planes introduced in [DGK14].
The last chapter describes a different construction which applies in a much
broader setting. The results in it are joint work with Nicolaus Treib. The key
notion that we use is that of a partial cyclic order on a set, introduced by Nova´k
in 1982 [Nov82]. They generalize cyclic orders in the same way that partial orders
generalize linear orders. The circle S1 has a cyclic ordering which is preserved by its
group of orientation-preserving homeomorphisms. Schottky groups can be defined
by choosing disjoint intervals and mapping them to the opposites of each other. In
particular, PSL(2,R) acts on the circle by orientation-preserving diffeomorphisms,
so real (or Fuchsian) Schottky groups are of this type. Section 1.1 explains this
motivating example. The construction for arbitrary partial cyclic orders is an analog
of this S1 example.
Under some topological assumptions about the partially ordered set, we show
that such Schottky groups admit left-continuous, equivariant limit curves. The
topological hypotheses are motivated by the main class of examples that we are in-
terested in : Shilov boundaries of Hermitian symmetric spaces. The cyclic nature of
3
these homogeneous spaces was previously observed and used to prove strong results
about discrete subgroups of Hermitian Lie groups ([Wie04], [BIW10], [BBH+16]).
We show the following:
Theorem 1.0.2. Let Σ be a compact, connected, orientable surface with non-empty
boundary and G a Lie group of Hermitian type. Let ρ : pi1(Σ) → G be a homo-
morphism. Then, ρ has maximal Toledo invariant if and only if there is a Schottky
presentation (in this cyclic sense) for the group ρ(pi1(Σ)) ⊂ G.
The Toledo invariant is a conjugacy invariant for representations of surface
groups into Lie groups of Hermitian type that was first introduced in 1979 [Tol79].
It generalizes the Euler number for representations of fundamental groups of closed
surfaces into PSL(2,R). Both invariants take values in a bounded range by the
Milnor-Wood inequality. Goldman [Gol80] showed that the Euler number assumes
its maximal value precisely for representations which correspond to holonomies of
hyperbolic structures. This motivated the study of representations which have max-
imal Toledo invariant and they geometric properties (see [BIW10]).
The concrete description in terms of Schottky groups that we use provides a
simple description of fundamental domains for the action of these representations.
For maximal representations into Sp(2n,R), we give a fundamental domain and
domain of discontinuity in RP2n−1. This domain of discontinuity had previously
been described in the Anosov case by Guichard and Wienhard [GW12], but not all
maximal representations are Anosov since they can contain unipotent elements.
Finally, an additional example of cyclic Schottky group is discussed, which is
4
inspired by the work of Fock and Goncharov on positivity of flags. For simplicity
we work with the space of oriented flags in R3, and we show that there is a partial
cyclic order on this space preserved by the action of the group PSL(3,R).
1.1 Fuchsian Schottky groups
Let V be a 2-dimensional vector space over R. Consider the projective line
P(V ). Its group of projective automorphisms is G = PGL(V ). In this section, we
describe a classical construction of free, discrete subgroups of G.
Definition 1.1.1. An interval I ⊂ P(V ) is an open interval in any affine patch
P(V )\{[v]}.
The complementary interval −I is defined to be the interval P(V )\I.
Let I±1 , . . . , I
±
k be 2k disjoint intervals in P(V ). Choose elements g1, . . . , gk ∈ G
satisfying the condition that gj(−I−j ) = I+j (see Figure 4.1). Then,
Lemma 1.1.2 (The Ping-pong lemma). The group Γ ⊂ G generated by g1, . . . , gk
is free on these generators.
We will actually prove the more general case:
Lemma 1.1.3. Let G be a group acting on a set X. Let γ1, . . . , γk ∈ G (k ≥ 2) be
elements of infinite order, and X1, . . . , Xk ⊂ X be non-empty, disjoint subsets such
that whenever i 6= j we have
γmi (Xj) ⊂ Xi.
for any m 6= 0. Then, the group generated by γ1, . . . , γk is free on those generators.
5
Proof. Consider an arbitrary reduced word w = γm1a1 . . . γ
m`
a`
in the generators γi
(here reduced just means ai 6= ai+1 and mi 6= 0). First, let’s assume a1 = a` and
choose any j 6= a`. We will look at the image of Xj under w.
wXj = γ
m1
a1
. . . γm`a` Xj ⊂ γm1a1 . . . γm`−1a`−1 Xa` ⊂ · · · ⊂ Xa1 .
By disjointness of the Xi we get that w acts nontrivially on X. For the remaining
cases (a1 6= a`), consider the word w′ = γm1a1 wγ−m1a1 = γ2m1a1 . . . γm`a` γm1a1 . By the
previous argument, w′ acts nontrivially, and so w also acts nontrivially.
Lemma 1.1.2 follows from this using γi = gi and Xi = I
+
i ∪ I−i .
If the defining intervals I±1 , . . . , I
±
k have disjoint closures, we call Γ a Fuchsian
Schottky group. These groups are intimately related to hyperbolic structures on
surfaces as follows.
Consider the upper half plane model {z ∈ C | =(z) > 0} of the hyperbolic
plane H2. Its boundary is naturally the real projective line RP1. Any projective au-
tomorphism of this boundary extends uniquely to an isometry of H2, and vice versa.
Any interval I ⊂ RP1 defines a unique open half plane in H2 by taking its convex
hull, and disjoint intervals correspond to disjoint half planes. A Fuchsian Schottky
group Γ thus acts on H2 by hyperbolic isometries. Additionally, the complement
in H2 of the half planes defining the Schottky group is a fundamental domain for
that action. The quotient is a hyperbolic surface with nonempty boundary, whose
topology is determined by the combinatorics of the Schottky construction.
6
Chapter 2: Lorentzian geometries
In this chapter we introduce three examples of constant curvature Lorentzian
3-manifolds. Then, we define the Einstein universe Ein which is the model for
conformal Lorentzian geometry. All three constant curvature examples conformally
embed in a natural way into the Einstein universe, and so they help understand
its geometry. In the next chapter, we study surfaces in Ein and their intersections
in order to build Schottky groups acting on it. This space is also one of the main
motivating examples for the general theory developed in Chapter 4.
2.1 Models for 3-dimensional Lorentzian space forms
A Lorentzian space form is a smooth manifold equipped with a constant sec-
tional curvature Lorentzian metric. There are three cases to consider : zero curva-
ture, negative curvature and positive curvature.
2.1.1 Minkowski space
Minkowski space is the flat model. It is analogous to Euclidean space in many
ways. For instance, it is homeomorphic to R3 and its geodesics are straight lines.
Let V 2,1 be a 3-dimensional real vector space and · be a nondegenerate sym-
7
metric bilinear form of signature (2, 1) on V 2,1. We call such a vector space a
Lorentzian vector space. Minkowski 3-space Min is an affine space with V 2,1 as its
vector space of translations.
The form · on V 2,1 defines at each point of Min a lightcone :
Definition 2.1.1. The lightcone of the point p ∈ Min is the set of points q ∈ Min
such that (q − p) · (q − p) = 0.
More precisely, · induces a trichotomy of vectors in V 2,1 which translates to
a trichotomy of points in Min whenever a base point is chosen. We call a vector
v ∈ V 2,1
• lightlike whenever v · v = 0,
• timelike whenever v · v < 0, and
• spacelike whenever v · v > 0
This classification of vectors into different types has consequences on every part of
the geometry of Minkowski space. For instance, geodesics have a type according
to their direction vector and this is preserved by any self isometries of Minkowski
space.
The timelike vectors in V 2,1 are divided into two connected components. Fixing
one of these components is called choosing a time orientation on V 2,1. We can use
this structure to define a partial order on Min. The timelike vectors in a fixed
component C form a proper convex cone. The partial order is then just defined by
q > p whenever q−p ∈ C. In this situation, we will sometimes say q is in the future
of p. This partial order will be crucial for the construction described in Chapter 4.
8
The isometry group of Minkowski space is the affine Lorentzian group V 2,1 o
O(V 2,1). Since V 2,1 is homogeneous space for a low dimensional Lie group, some
isomorphisms between Lie algebras give different models of this space.
The Lie algebra isomorphism o(V 2,1) ∼= sl(2,R) has the useful application that
the Lie algebra sl(2,R) of traceless 2 × 2 matrices itself is a model for Minkowski
space. We now describe this model.
Definition 2.1.2. The Killing form on sl(2,R) is the bilinear form K(X, Y ) =
Tr(XY )/2.
The basis 1 0
0 −1
 ,
0 1
1 0
 ,
 0 1
−1 0

is orthonormal for K, and this shows that this symmetric bilinear form is nonde-
generate and has signature (2, 1). Hence, this Lie algebra inherits the geometry of
Minkowski space. The classification of vectors reflects the classification of isome-
tries of the hyperbolic plane. The timelike vectors correspond to infinitesimal elliptic
isometries, the lightlike vectors to infinitesimal parabolic isometries, and the space-
like vectors to infinitesimal hyperbolic isometries.
The group of orientation preserving, time preserving isometries of Min in this
model is PSL(2,R)n sl(2,R), acting on sl(2,R) in the following way:
(A, Y ) ·X := Ad(A)X + Y.
9
2.1.2 Anti de Sitter space
Anti de Sitter space is the negatively curved model. It is analogous to hyper-
bolic space.
Let V 2,2 be a vector space of dimension 4 over R and · a symmetric bilinear
form of signature (2, 2) on V 2,2.
Definition 2.1.3. Anti de Sitter space is the following submanifold of V 2,2:
AdS := {v ∈ V 2,2 | v · v = −1}.
The ambient pseudo-Riemannian metric of signature (2, 2) restricts to a metric
of signature (2, 1) on the tangent spaces of this submanifold. The isometry group
of AdS is O(V 2,2).
In this space, the spacelike and lightlike geodesics are all infinite, but the
timelike geodesics are all closed.
Once again, low dimensional “accidental” isomorphisms between Lie algebras
give different models of this space. The vector space of all 2 × 2 matrices over R
has a natural quadratic form of signature (2, 2) : the determinant. If we fix the
symmetric bilinear form associated to − det instead, we get that the submanifold
AdS is just the group SL(2,R).
There is an equivariant isomorphism between the actions of the identity com-
ponent of O(V 2,2) on AdS and the identity component of SL(2,R) × SL(2,R) on
SL(2,R) by left and right multiplication. This is due to the isomorphism o(2, 2) ∼=
sl(2,R)⊕ sl(2,R).
10
2.1.3 de Sitter space
The third and last model for constant curvature Lorentzian 3-manifolds is de
Sitter space. It is positively curved, and analogous to the Riemannian 3-sphere. It
is defined in a similar way to anti de Sitter space : Let V 3,1 be a vector space with
a bilinear form · of signature (3, 1).
Definition 2.1.4. de Sitter space is the submanifold
dS := {v ∈ V 3,1 | v · v = 1}.
Note that if we had set the condition on the right to v ·v = −1, we would have
obtained a Riemannian manifold : hyperbolic 3-space. This relationship induces a
duality between points in dS and totally geodesic planes in H3 coming from orthog-
onality in V 3,1. Similarly, points in H3 are dual to totally geodesic spacelike planes
in dS.
2.2 The Einstein universe
In this section we will describe the model geometry for conformally flat Lorentzian
3-manifolds.
First, let’s recall some facts about the conformal Riemannian sphere. The
three constant curvature Riemannian manifolds admit conformal embeddings into
the sphere. The flat Euclidean space embeds as the complement of a point using
stereographic projection. The negatively curved hyperbolic space embeds as a hemi-
sphere (or, alternatively, a disjoint union of two hyperbolic spaces embeds as the
11
complement the equatorial sphere). Finally, the positively curved sphere embeds
as the whole conformal sphere. We will see that each of these embeddings has an
analog in the Lorentzian setting.
2.2.1 The projective model
The first model of the 3-dimensional Einstein universe that we will discuss is
the projective model. It sits as a submanifold of projective 4-space.
Let V 3,2 be a 5-dimensional real vector space endowed with a signature (3, 2)
symmetric bilinear form.
Definition 2.2.1. The Einstein universe is the submanifold of P(V 3,2) defined by
Ein := {[v] ∈ P(V 3,2) | v · v = 0}
In order to make the conformal metric more explicit, we can choose a (positive
definite) scalar product 〈, 〉 on V 3,2 and look at the double cover
E˜in := {v ∈ V 3,2 | v · v = 0 and 〈v, v〉 = 1}
Each point of Ein has exactly two representatives in this submanifold of V 3,2. More-
over, the ambient metric restricts to a Lorentzian metric on E˜in invariant under the
antipodal map. Finally, choosing any other local section of the projection P will
change this metric by a conformal map.
The group of conformal automorphisms of Ein is the orthogonal group O(V 3,2).
In order to conformally embed Minkowski space into the Einstein universe,
first choose any (2, 1) subspace V 2,1 ⊂ V 3,2. Now, (V 2,1)⊥ has signature (1, 1),
12
which means that there are exactly two projective equivalences of null vectors in it.
Choose representatives p, q for these two null directions, normalized so that p ·q = 1.
The map E : V 2,1 → V 3,2 defined by v 7→ q − v·v
2
p + v embeds V 2,1 into the
null cone of V 3,2.
Similarly, in order to conformally embed anti de Sitter (respectively de Sitter)
space into the Einstein universe, fix a signature (2, 2) (respectively (3, 1)) subspace
U of V 3,2. Choose one of the two vectors v ∈ V 3,2 orthogonal to U such that v ·v = 1
(respectively v · v = −1). Define E : U → V 3,2 by E(u) = u + v. When restricted
to AdS ⊂ U (respectively dS ⊂ U) the image of E is in the lightcone of V 3,2.
The easiest way to show that these maps are conformal embeddings is to see
that they are equivariant with respect to the transitive isometry groups of each of
the spaces in question.
2.2.2 Geometric objects
The natural objects to study in Euclidean geometry are straight lines, circles
and triangles. In this section, we describe classes of curves and surfaces in the
Einstein universe that are natural to study.
The first of these objects is the photon, or lightlike geodesic. This is the
only type of pseudo-Riemannian geodesic which is invariant under all conformal
transformations.
Definition 2.2.2. A linear subspace is called isotropic if it is contained in its or-
thogonal subspace. A photon in Ein is the projectivization of an isotropic 2-plane
13
in V 3,2.
The conformal group O(V 3,2) acts transitively on photons. We will denote the
homogeneous space of photons by Pho.
Photons together with points define an incidence relation. We say that two
points p, q ∈ Ein are incident whenever there is a photon ϕ ∈ Pho containing both
: p, q ∈ ϕ. In this projective model, this is equivalent to the two isotropic lines
defining p and q being orthogonal with respect to ·. Similarly, two photons are said
to be incident whenever they intersect in a point. Finally, a point is incident to a
photon if it is contained in that photon.
Lemma 2.2.3. Let p ∈ Ein and ϕ ∈ Pho with p not incident to ϕ. Then, there is
a unique photon ψ incident to both p and ϕ, and a unique point q also incident to
both p and ϕ.
Proof. Assume p = [u] and ϕ = 〈v, w〉. We first show the incidence of q. Define
q = [(u · v)w − (u · w)v] ∈ ϕ. Now,
((u · v)w − (u · w)v) · u = 0,
so q is incident to p. If there existed another q′ with these properties, we would have
that p, q, q′ are pairwise incident and distinct, which would imply that there is an
isotropic 3-dimensional subspace of V 3,2, a contradiction.
For the rest of the lemma, notice that ψ = 〈p, q〉 is incident to both p and ϕ,
and uniqueness of q proves uniqueness of ψ.
Definition 2.2.4. The lightcone of a point [v] ∈ Ein is the set of all points incident
14
to [v]. Equivalently, it is the union of all photons containing [v].
L([v]) = {[u] ∈ Ein|u · v = 0}
Different types of subspaces in V 3,2 define other natural curves and surfaces.
Definition 2.2.5. A timelike (respectively spacelike) circle is the projectivization
of the nullcone of a signature (1, 2) (respectively (2, 1)) subspace in V 3,2.
Proposition 2.2.6. Let p, q ∈ Ein be a pair of non-incident points. Then, L(p) ∩
L(q) is a spacelike circle. Conversely, for each spacelike circle S, there is a unique
pair of points p, q with L(p) ∩ L(q) = S.
Proof. Since p, q are non-incident, they span a (1, 1) subspace. The intersection of
their lightcones is the projectivized lightcone of the orthogonal to that subspace,
which has signature (2, 1).
For the converse, consider the orthogonal complement of the (2, 1) subspace
defining the spacelike circle. It has signature (1, 1) and so contains exactly two
lightlike directions, corresponding to the points p, q.
Proposition 2.2.7. Three pairwise non-incident points p, q, r ∈ Ein define a unique
timelike circle or spacelike circle going through them.
Proof. By the non-incidence property, the span of p, q, r is a nondegenerate three-
dimensional subspace. It cannot be positive definite since it contains null lines p, q, r.
Hence, it has to be either a signature (2, 1) or a signature (1, 2) subspace.
The previous proposition allows us to define a relation on triples of pairwise
non-incident points in the Einstein universe.
15
Definition 2.2.8. We call a triple of pairwise non-incident points p, q, r in the
Einstein universe a :
• timelike triple if there is a timelike circle through them, or a
• spacelike triple if there is a spacelike circle through them
Definition 2.2.9. An Einstein torus is the projectivization of the nullcone of a
signature (2, 2) subspace in V 3,2. It is an embedded copy of the 2-dimensional
Einstein universe.
Definition 2.2.10. A Riemann sphere is the projectivization of the nullcone of a
signature (3, 1) subspace in V 3,2. It is an embedded copy of the conformal 2-sphere.
2.2.3 The Lagrangian Grassmannian model
In Sections 2.1.1, 2.1.2 and 2.1.3, we discussed alternate models for each of the
constant curvature models using low dimensional Lie group isomorphisms. Conve-
niently, there is also a low dimensional isomorphism giving an alternate model for
the Einstein universe. It is the isomorphism of Lie groups SO0(3, 2) ∼= PSp(4,R).
Let V be a 4-dimensional vector space over R. Equip V with a nondegenerate,
skew-symmetric bilinear form ω, making it into a symplectic vector space. The Lie
group preserving this structure is Sp(4,R). A 2-dimensional subspace of V is called
Lagrangian if the restriction of ω to the subspace is identically zero. The Grassman-
nian of 2-planes in V is the space of 2-dimensional subspaces of V . The subspace
consisting of only the Lagrangian subspaces is called the Lagrangian Grassmannian.
This is the model of the Einstein universe that we will describe in this section.
16
In this model, points correspond to Lagrangian planes and photons correspond
to lines in V . A point is incident to a photon when the line corresponding to
the photon is contained in the Lagrangian plane corresponding to the point. Two
points are incident when the corresponding Lagrangians intersect, and finally two
photons are incident when the corresponding lines span a Lagrangian (see Section
3.2 and [BCD+08] for details).
Minkowski patches in this model are most easily described in terms of the
sl(2,R) model of Minkowski space. Let L,L′ ⊂ V be a pair of transverse 2-
dimensional subspaces of V .
Definition 2.2.11. Let f : L → L′ be a linear map. The graph of f is the linear
subspace
graph(f) := {v + f(v) | v ∈ L}
Next, assume L,L′ are transverse Lagrangians, so they correspond to a pair
of non-incident points in the Einstein universe.
Lemma 2.2.12. The graph of f : L → L′ is a Lagrangian subspace if and only if
ω(u, f(v)) + ω(f(u), v) = 0 for all u, v ∈ L.
Proof. Let u, v ∈ L. Then,
ω(u+ f(u), v + f(v)) = ω(u, f(v)) + ω(f(u), v).
Let σ ∈ Sp(V, ω) be an involution such that σ(L) = L′. This defines a sym-
plectic form on L as follows: ωσ(u, v) := ω(u, σ(v)). Then, the condition of Lemma
17
2.2.12 is exactly the condition that σ ◦ f ∈ sp(L, ωσ). Therefore, we get a map
sp(L, ωσ)→ Ein
σ ◦ f 7→ graph(f).
Since sp(L, ωσ) ∼= sl(2,R), Minkowski space embeds into this model of the Einstein
universe. Explicitly, let us use the standard symplectic form given by the block
matrix  0 I
−I 0
 ,
and the involution
σ =

0 0 0 1
0 0 −1 0
0 −1 0 0
1 0 0 0

∈ Sp(4,R)
which interchanges the Lagrangians L and L′ spanned respectively by the first two
basis vectors and the last two basis vectors. Then, we find that ωσ is given by the
matrix 0 −1
1 0
 .
Thus, with these choices the Lie algebra sp(L, ωσ) consist of the standard traceless
2× 2 matrices, and the map giving the embedding is
a b
c −a
 7→

1 0
0 1
−c a
a b

18
where the 4×2 matrix on the right represents the Lagrangian spanned by its columns.
We see that with these conventions, the Minkowski patch coincides with graphs of
linear maps given by a symmetric matrix. Moreover, the Lorentzian quadratic form
is given by minus the determinant for both the traceless and the symmetric matrix.
Let us now describe the embedding of anti de Sitter space using the SL(2,R)
model in an analogous way.
Let S be a non-Lagrangian plane in V . Then, S⊥ is also non-Lagrangian and
V = S ⊕ S⊥.
Proposition 2.2.13. The graph of a linear map f : S → S⊥ is Lagrangian if and
only if ω(f(u), f(v)) = −ω(u, v) for all u, v ∈ S.
Proof.
ω(u+ f(u), v + f(v)) = ω(u, v) + ω(f(u), f(v))
Let i be an antisymplectic isomorphism between S and S⊥. This means that
ω(i(u), i(v)) = −ω(u, v) for all u, v ∈ S. Then, the condition above translates to
the condition that i−1 ◦ f preserves ω on S, that is, i−1 ◦ f ∈ Sp(S, ω). This defines
a map
Sp(S, ω)→ Ein
i−1 ◦ f 7→ graph(f).
19
Since Sp(S, ω) ∼= SL(2,R), this defines an embedding of anti de Sitter space into
this model of the Einstein universe.
In order to describe the embedding of de Sitter space into the Einstein universe,
we will do something slightly different. We will use the fact that, as a homogeneous
space, dS ∼= SL(2,C)/SL(2,R).
Let ωC be a complex-valued, skew-symmetric nondegenerate bilinear form on
C2. Then, its imaginary part defines a real valued symplectic form on C2. The group
SL(2,C) ∼= Sp(2,C) of symplectic linear transformations in particular preserves the
imaginary part, so we get an injective homomorphism SL(2,C) → Sp(4,R). The
stabilizer of a real 2-plane in C which is not a complex line is isomorphic to SL(2,R),
realizing dS as a subspace of the real Lagrangian Grassmannian.
We now describe some of the geometric objects from Section 2.2.2 in this
model. For this purpose, we need to define the Maslov index of a triple of La-
grangians.
Let P,Q,R be three pairwise transverse Lagrangians in V . Denote by piP , piR
the projections associted to the splitting V = P ⊕R.
Definition 2.2.14. The Maslov index of the triple P,Q,R, denoted m(P,Q,R) is
the signature of the following nondegenerate quadratic form on Q :
BP,R(u) = ω(piP (u), piR(u)).
Here, signature means the difference between the number of positive and negative
eigenvalues of BP,R.
Proposition 2.2.15. The Maslov index enjoys the following properties for all P,Q,R, S
20
pairwise transverse Lagrangians
• m(P,Q,R) ∈ {−2, 0, 2}
• m(P,Q,R) = m(AP,AQ,AR) for any A ∈ Sp(4,R)
• m(P,Q,R) = m(Q,R, P )
• m(P,Q,R)−m(Q,R, S) +m(P,R, S)−m(Q,R, S) = 0.
The invariant m(P,Q,R) distinguishes triples on pairwise non-incident points
according to their type.
Proposition 2.2.16. Let P,Q,R be a triple of pairwise transverse Lagrangians.
They form a timelike triple when m(P,Q,R) = ±2 and a spacelike triple when
m(P,Q,R) = 0 (see Definition 2.2.8).
Proof. Let e1, e2, e3, e4 be a symplectic basis of V . Using the action of Sp(4,R), we
can assume that P is spanned by e1, e2 and R is spanned by e3, e4. By transversality,
we can writeQ as the graph of a unique linear map f : P → R. In the bases e1, e2 and
e3, e4, f will be represented by a symmetric matrix F . The quadratic form BP,R is
also represented by F , and the Lorentzian quadratic form in this model of Minkowski
space is minus the determinant. Thus, when det(F ) < 0, the triple is spacelike, the
quadratic form is indefinite and so m(P,Q,R) = 0. Similarly, when det(F ) > 0 the
triple is timelike and the quadratic form is definite so m(P,Q,R) = ±2.
Proposition 2.2.17. Let L,L′ be transverse Lagrangians. The set of all Lagrangians
intersecting both L and L′ is a spacelike circle. Conversely, all spacelike circles are
of this form.
21
Proof. By Proposition 2.2.6, intersections of two lightcones and spacelike circles are
equivalent. In the Lagrangian model, the lightcone of a Lagrangian is the set of a
Lagrangians intersecting it, so the result is immediate.
Proposition 2.2.18. Let S be a nondegenerate 2-dimensional subspace of V (that is,
the symplectic form ω does not vanish on S). The set of all Lagrangians intersecting
S is an Einstein torus. The subspace S is uniquely determined by the spacelike circle,
up to replacing S by its symplectic orthogonal complement S⊥.
We postpone the proof of this last statement to Chapter 3 where we investigate
Einstein tori in detail (more precisely Section 3.2.1 for this proposition).
2.2.4 The Lie circles model
We show in this section that the Einstein universe is the moduli space of ori-
ented circles in the 2-sphere. Cecil [Cec08] explains this through the usual projective
model of the Einstein universe. However, as far as the author knows, there is no
exposition of this using the Lagrangian Grassmannian model. We will develop the
theory from this point of view.
Let V = C2 considered as a 4-dimensional real vector space. The determinant
provides a complex symplectic form on C2, and both its real and imaginary parts
are real symplectic forms. We will use the form ω := =(det). A Lagrangian for this
form is a real 2-dimensional subspace of C2 which is spanned by two vectors with a
real determinant.
Proposition 2.2.19. The projectivization of a real 2-plane P ⊂ C2 is a circle or a
22
point in CP1.
Proof. Let u, v be vectors whose R-span is P . Denoting the projection to CP1 by pi,
pi(P ) = {pi(ku+ lv) | k, l ∈ R} =

u1 v1
u2 v2

k
l

∣∣∣∣∣ k, l ∈ R
 .
If the matrix with columns u, v is singular over C, then the image is a single point.
Otherwise, it is the image of a circle (the extended real line) by a mo¨bius transfor-
mation, so is a circle.
Fix a circle C ⊂ CP1 and a pair of distinct points [p], [q] ∈ C. Then, any
pair of representatives for [p], [q] will define a real plane in C2 which projects to C.
Since real changes of basis do not affect the plane, the collection of planes which
project to C is the set {eiθ(Rp+Rq), θ ∈ [0, pi)}. The Lagrangian planes in this set
must have =(det(eiθp, eiθq)) = 0. This equation has exactly two solutions which are
interchanged by multiplication by i. We conclude:
Proposition 2.2.20. The collection of real Lagrangian 2-planes in C2 which are
not complex lines projects 2-to-1 to the collection of circles in CP1.
Since multiplication by i preserves complex lines and switches the two real
planes projecting to any circle, we can interpret this as a change of orientation on
the set of circles. This means that Lagrangians in C2 correspond to oriented circles
and points in CP1.
Proposition 2.2.21. Assume L,L′ ⊂ C2 are a pair of real Lagrangian 2-planes
which are not complex lines. Assume moreover that L and L′ intersect in a line `.
Then, pi(L) and pi(L′) correspond to tangent circles in the plane.
23
Proof. Let u = (u1, u2) ∈ ` and assume without loss of generality that u2 6= 0.
Complete u to a basis of L and L′ with vectors v, v′, respectively. Then, in a
neighborhood of pi(u), the projections can be parameterized by
γ(t) =
u1 + v1t
u2 + v2t
and
γ′(t) =
u1 + v
′
1t
u2 + v′2t
.
The tangent vectors to these paths at pi(u) = γ(0) = γ′(0) are given respectively by
γ˙(0) =
u1v2 − u2v1
u22
γ˙′(0) =
u1v
′
2 − u2v′1
u22
.
To show that the circles are tangent, it remains to show that these vectors are real
multiples of each other. The quotient of the two values is
γ
γ′
=
u1v2 − u2v1
u1v′2 − u2v′1
,
which has a real numerator and a real denominator since we assumed that L and L′
were Lagrangian.
More precisely, L and L′ intersect in a line exactly when the oriented circles
they correspond to are tangent with matching orientations at the tangency point.
24
Figure 2.1: A photon in the circles model of the Einstein universe.
25
Figure 2.2: A spacelike circle in the circles model. The orientations on the outer
and inner circle are opposite, and the orientations of all the other circles match that
of the outer circle.
26
Chapter 3: Crooked Schottky groups
The content of this chapter is essentially from the preprint [BCFG17]. We
want to use a class of hypersurfaces in the Einstein universe called crooked surfaces
in order to build fundamental domains for Schottky groups. Since these surfaces are
defined piecewise in a non-trivial way, the difficulty lies in finding a configuration
of such surfaces which are disjoint. In order to find a disjointness criterion, we first
focus on Einstein tori (Definition 2.2.9) and describe their intersections.
3.1 Einstein tori
The purpose of this section is to define an invariant η ≥ 0 characterizing
pairs of Einstein tori in Ein. Then, we interpret this invariant in the Lagrangian
Grassmannian model. Let V 3,2 be a real vector space of dimension 5 endowed with
a signature (3, 2) symmetric bilinear form ·.
3.1.1 Pairs of positive vectors
If s ∈ V 3,2 is spacelike, then s⊥ is a subspace of signature (2, 2), which means
that its projectivized lightcone is an Einstein torus.
A linearly independent pair of two unit-spacelike vectors s1, s2 spans a 2-plane
27
〈s1, s2〉 ⊂ V 3,2 which is:
• Positive definite ⇐⇒ |s1 · s2| < 1;
• Degenerate ⇐⇒ |s1 · s2| = 1;
• Indefinite ⇐⇒ |s1 · s2| > 1.
The positive definite and indefinite cases respectively determine orthogonal splittings
V 3,2 ∼= R3,2 = R2,0 ⊕ R1,2
V 3,2 ∼= R3,2 = R1,1 ⊕ R2,1.
In the degenerate case, the null space is spanned by s1 ± s2, where
s1 · s2 = ∓1.
By replacing s2 by −s2 if necessary, we may assume that s1 · s2 = 1. Then s1 − s2
is null. Since V 3,2 itself is nondegenerate, there exists v3 ∈ V 3,2 such that
(s1 − s2) · v3 = 1.
Then s1, s2, v3 span a nondegenerate 3-dimensional subspace of signature (2, 1).
Let H1, H2 be the Einstein tori respectively defined by the orthogonal sub-
spaces s⊥1 , s
⊥
2 . The absolute value of the product
η(H1, H2) := |s1 · s2|
is a nonnegative real number, depending only on the pair of Einstein hyperplanes
H1 and H2. We have thus proved the following classification for pairs of Einstein
tori:
28
• If the span of s1, s2 is positive definite (η(H1, H2) < 1), then the intersection
of the corresponding Einstein tori is the projectivised null cone of a signature
(1, 2) subspace, which is a timelike circle.
• If the span of s1, s2 is indefinite (η(H1, H2) > 1), then the intersection is the
projectivised null cone of a signature (2, 1) subspace, which is a spacelike circle.
• Finally, if the span of s1, s2 is degenerate (η(H1, H2) = 1), then the inter-
section is the projectivised null cone of a degenerate subspace with signature
(+,−, 0). This null cone is exactly the union of two isotropic planes intersect-
ing in the degenerate direction, so when projectivising we get a pair of photons
intersecting in a point.
Corollary 3.1.1. The intersection of two Einstein tori is noncontractible in each
of the two tori.
Proof. An Einstein torus is a copy of the 2-dimensional Einstein universe. Explicitly,
we can write it as P(N ) where N is the null cone in R2,2. A computation shows that
all timelike circles are homotopic, all spacelike circles are homotopic and these two
homotopy classes together generate the fundamental group of the torus. Similarly,
photons are homotopic to the sum of these generators and so are noncontractible.
3.1.2 Involutions in Einstein tori
Orthogonal reflection in s defines an involution of Ein which fixes the corre-
sponding hyperplane H = s⊥. The orthogonal reflection in a positive vector s is
29
(a) Two photons (b) A timelike circle (c) A spacelike circle
Figure 3.1: The three possible types of intersection for a pair of Einstein tori, viewed
in a Minkowski patch.
defined by:
Rs(v) = v − 2v · s
s · ss.
We compute the eigenvalues of the composition RsRs′ , where s, s
′ are unit spacelike
vectors, and relate this to the invariant η.
The orthogonal subspace to the plane spanned by s and s′ is fixed pointwise
by this composition. Therefore, 1 is an eigenvalue of multiplicity 3. In order to
determine the remaining eigenvalues, we compute the restriction of RsRs′ to the
subspace Rs+ Rs′.
30
RsRs′(s) = Rs(s− 2(s · s′)s′)
= −s− 2(s · s′)(s′ − 2(s′ · s)s)
= (4(s′ · s)2 − 1)s− 2(s′ · s)s′.
RsRs′(s
′) = Rs(−s′)
= −s′ + 2(s · s′)s.
The matrix representation of RsRs′ in the basis s, s
′ is therefore:4(s′ · s)2 − 1 2(s · s′)
−2(s′ · s) −1
 .
The eigenvalues of this matrix are:
2(s · s′)2 − 1± 2(s · s′)
√
(s · s′)2 − 1.
We observe that they only depend on the invariant η = |s · s′|. The composition of
involutions has real distinct eigenvalues when the intersection is spacelike, complex
eigenvalues when the intersection is timelike, and a double real eigenvalue when the
intersection is a pair of photons.
The case when s1 · s2 = 0 is special: in that case the two involutions commute
and we will say that the Einstein tori are orthogonal. The complement of an Einstein
torus in Ein is a model for the double covering space of anti de Sitter space AdS3.
In this conformal model of AdS3 (see [Gol15]), indefinite totally geodesic 2-planes
are represented by tori which are orthogonal to the Einstein torus ∂AdS3.
31
3.2 The Lagrangian Grassmannian model
We first recall the Lagrangian model of the Einstein universe described in Sec-
tion 2.2.3. Then, we make the identification between this model and the projective
model more explicit. We describe Einstein tori in this context and the invariant η
for a pair of tori.
Let (V, ω) be a 4-dimensional real symplectic vector space, that is, V is a real
vector space of dimension 4 and V × V ω−−→ R is a nondegenerate skew-symmetric
bilinear form. Let vol ∈ Λ4(V ) be the element defined by the equation (ω∧ω)(vol) =
−2. The second exterior power Λ2(V ) admits a nondegenerate symmetric bilinear
form · of signature (3, 3) defined by
(u ∧ v) ∧ (u′ ∧ v′) = (u ∧ v) · (u′ ∧ v′)vol.
The kernel
W := Ker(ω) ⊂ Λ2(V )
inherits a symmetric bilinear form which has signature (3, 2).
Define the vector ω∗ ∈ Λ2V to be dual to ω by the equation
ω∗ · (u ∧ v) = ω(u, v),
for all u, v ∈ V . Because of our previous choice of vol, we have ω∗ · ω∗ = −2. The
bilinear form ·, together with the vector ω∗ define a reflection
Rω∗ : Λ
2(V )→ Λ2(V )
u 7→ u+ (u · ω∗)ω∗.
32
The fixed set of this reflection is exactly the vector subspace W orthogonal to ω∗.
The Plu¨cker embedding ι : Gr(2, V ) → P(Λ2(V )) maps 2-planes in V to lines
in Λ2(V ). We say that a plane in V is Lagrangian if the form ω vanishes identically
on pairs of vectors in that plane. If we restrict ι to Lagrangian planes, then the
image is exactly the set of null lines in W .
The form ω yields a relation of orthogonality on 2-planes in V . Lagrangian
planes are orthogonal to themselves, and non-Lagrangian planes have a unique or-
thogonal complement which is also non-Lagrangian. The following proposition re-
lates orthogonality in V with a reflection operation on Λ2(V ).
Proposition 3.2.1. A pair of 2-dimensional subspaces S, T ⊂ V are orthogonal
with respect to ω if and only if [Rω∗(ι(S))] = [ι(T )].
Proof. First, assume S is Lagrangian. This means that S = S⊥, and that ι(S) ∈
ω∗⊥. Hence,
Rω∗(ι(S)) = ι(S) = ι(S
⊥).
Next, if S is not Lagrangian, then we can find bases (u, v) of S and (u′, v′) of
S⊥ satisfying ω(u, v) = ω(u′, v′) = 1 and all other products between these four are
zero. Then,
vol = −u ∧ v ∧ u′ ∧ v′
and
ω∗ = −u ∧ v − u′ ∧ v′.
Consequently,
[Rω∗(ι(S))] = [u ∧ v + ω(u, v)ω∗] = [−u′ ∧ v′] = [ι(S⊥)].
33
Recall that a point and a photon are called incident if the point is in the
photon (Section 2.2.2). This incidence relation is reflected in the two models in the
following way: A point p ∈ Ein and a photon φ ∈ Pho are incident if and only if
(p, φ) satisfies one of the two equivalent conditions:
• The null line in W corresponding to p lies in the isotropic 2-plane in W cor-
responding to φ.
• The Lagrangian 2-plane in V corresponding to p contains the line in W cor-
responding to φ.
The following proposition proves this equivalence :
Proposition 3.2.2. Let P,Q ⊂ V be two-dimensional subspaces. Then, P ∩Q = 0
if and only if ι(P ) · ι(Q) 6= 0.
Proof. Choose bases u, v of P and u′, v′ of Q. Then,
u ∧ v ∧ u′ ∧ v′ 6= 0
if and only if u, v, u′, v′ span V which is equivalent to P and Q being transverse.
The light cone of p corresponds the orthogonal hyperplane [p]⊥ ⊂ W of the
null line corresponding to p. In photon space P(V ), the photons containing p form
the projective space P(L) of the Lagrangian 2-plane L corresponding to p.
34
3.2.1 Nondegenerate planes and symplectic splittings
We describe the algebraic structures equivalent to an Einstein torus in Ein. As
a reminder, these are hyperplanes of signature (2, 2) inside W ∼= R3,2, and describe
surfaces in Ein homeomorphic to a 2-torus.
In symplectic terms, an Einstein torus corresponds to a splitting of V as a
symplectic direct sum of two nondegenerate 2-planes. Let us detail this correspon-
dence.
Define a 2-dimensional subspace S ⊂ V to be nondegenerate if and only if
the restriction ω|S is nondegenerate. A nondegenerate 2-plane S ⊂ V determines a
splitting as follows. The plane
S⊥ := {v ∈ V | ω(v, S) = 0}
is also nondegenerate, and defines a symplectic complement to S. In other words,
V splits as an (internal) symplectic direct sum:
V = S ⊕ S⊥.
The corresponding Einstein torus is then the set of Lagrangians which are non-
transverse to S (and therefore also to S⊥).
The lines in S determine a projective line in Pho which is not Legendrian.
Conversely, non-Legendrian projective lines in Pho correspond to nondegenerate 2-
planes. This non-Legendrian line in Pho, as a set of photons, corresponds to one
of the two rulings of the Einstein torus. The other ruling corresponds to the line
P(S⊥).
35
In order to make explicit the relationship between the descriptions of Einstein
tori in the two models, define a map µ as follows:
µ : Gr(2, V )→ W
S 7→ ι(S) + 1
2
ω(ι(S))ω∗.
This is the composition of the Plucker embedding ι with the orthogonal projection
onto W .
Lemma 3.2.3. For S a nondegenerate plane, the image of µ is always a spacelike
vector, and µ(S) = µ(S⊥).
Proof. For the first part,
µ(S) · µ(S) = 1
2
ω(ι(S))2 > 0.
The second part is a consequence of the correspondence between orthogonal com-
plements and reflection in ω∗ (Proposition 3.2.1) and the fact that a vector and
its reflected copy have the same orthogonal projection to the hyperplane of reflec-
tion.
Proposition 3.2.4. The map µ induces a bijection between spacelike lines in W
and symplectic splittings of V . Under the Plu¨cker embedding ι, the Einstein torus
defined by the symplectic splitting S ⊕ S⊥ is sent to the Einstein torus defined by
the spacelike vector µ(S) ∈ W .
Proof. Let u ∈ W be a spacelike vector normalized so that u · u = 2. Then, both
vectors u ± ω∗ are null. By the fact that null vectors in Λ2(V ) are decomposable,
36
each u± ω∗ corresponds to a 2-plane in V . These 2-planes are nondegenerate since
(u± ω∗) ∧ ω∗ = −ω(u± ω∗)vol = 2 6= 0.
The two planes u±ω∗ are orthogonal since they are the images of each other by the
reflection Rω∗ , and so they are the summands for a symplectic splitting of V .
The map associating to u the splitting u ± ω∗ is inverse to the projection µ
defined above.
To prove the last statement in the proposition, we apply Proposition 3.2.2.
The Einstein torus defined by the splitting S, S⊥ is the set of Lagrangian planes
which intersect S (and S⊥) in a nonzero subspace. Let P be such a plane. Then,
ι(S) · ι(P ) = 0, which means that
(
ι(S) +
1
2
(ι(S) · ω∗)ω∗
)
· ι(P ) = 0,
so ι(P ) is in the Einstein torus defined by the orthogonal projection of S. Similarly,
if ι(P ) is orthogonal to uS then P intersects S in a nonzero subspace.
3.2.2 Graphs of linear maps
Now we describe pairs Einstein tori in terms of symplectic splittings of (V, ω)
more explicitly.
Let A,B be vector spaces of the same dimension and A⊕B their direct sum. If
A
f−→ B is a linear map, then the graph of f is the linear subspace graph(f) ⊂ A⊕B
consisting of all a ⊕ f(a), where a ∈ A. Every linear subspace L ⊂ A ⊕ B which
is transverse to B = 0 ⊕ B ⊂ A ⊕ B and having the same dimension as A, equals
37
graph(f) for a unique f . Furthermore, L = graph(f) is transverse to A = A ⊕ 0
if and only if f is invertible, in which case L = graph(f−1) for the inverse map
B
f−1−−→ A.
Suppose that A,B are vector spaces with nondegenerate alternating bilinear
forms ωA, ωB, respectively. Let A
f−→ B be a linear map. Its adjugate is the linear
map
B
Adj(f)−−−→ A
defined as the composition
B
ω#B−−→ B∗ f†−→ A∗ ω
#
A−−→ A (3.1)
where ω#A , ω
#
B are isomorphisms induced by ωA, ωB respectively, and f
† is the trans-
pose of f . If a1, a2 and b1, b2 are bases of A and B respectively with
ωA(a1, a2) = 1
ωB(b1, b2) = 1,
then the matrices representing f and Adj(f) in these bases are related by:
Adj
f11 f12
f21 f22
 =
 f22 −f12
−f21 f11
 .
In particular, if f is invertible and dim(A) = dim(B) = 2, then
Adj(f) = Det(f)f−1
where Det(f) is defined by f ∗(ωB) = Det(f)ωA.
Lemma 3.2.5. Let V = S ⊕ S⊥. Let S f−→ S⊥ be a linear map and let P =
graph(f) ⊂ V be the corresponding 2-plane in V which is transverse to S⊥.
38
• P is nondegenerate if and only if Det(f) 6= −1.
• If P is nondegenerate, then its complement P⊥ is transverse to S, and equals
the graph
P⊥ = graph
(− Adj(f)),
of the negative of the adjugate map to f
S⊥
−Adj(f)−−−−−→ S.
Proof. Choose a basis a, b for S. Then a ⊕ f(a) and b ⊕ f(b) define a basis for P ,
and
ω
(
a⊕ f(a), b⊕ f(b)) = ω(a, b)+ ω(f(a), f(b))
=
(
1 + Det(f)
)
ω
(
a, b
)
,
since, by definition,
ω
(
f(a), f(b)
)
= Det(f)ω
(
a, b
)
.
Thus P is nondegenerate if and only if 1 + Det(f) 6= 0, as desired.
For the second assertion, suppose that P is nondegenerate. Since P, P⊥, S, S⊥ ⊂
V are each 2-dimensional, the following conditions are equivalent:
• P is transverse to S⊥;
• P ∩ S⊥ = 0;
• P⊥ + S = V ;
• P⊥ is transverse to S.
Thus P⊥ = graph(g) for a linear map S⊥
g−→ S.
39
We express the condition that ω(P, P⊥) = 0 in terms of f and g: For s ∈ S
and t ∈ S⊥, the symplectic product is zero if anly only if
ω
(
s+ f(s), t+ g(t)
)
= ω
(
s, g(t)
)
+ ω
(
f(s), t
)
(3.2)
vanishes. This condition easily implies that g = −Adj(f) as claimed.
The following proposition relates the invariant η defined for a pair of spacelike
vectors with the invariant Det associated to a pair of symplectic splittings.
Proposition 3.2.6. Let S ⊕ S⊥ be a symplectic splitting and f : S → S⊥ be a
linear map with Det(f) 6= −1. Let T = graph(f) be the symplectic plane defined by
f . Then,
η(µ(S), µ(T )) =
|µ(S) · µ(T )|√
(µ(S) · µ(S))(µ(T ) · µ(T )) =
|1− Det(f)|
|1 + Det(f)| .
Proof. Let u, v be a basis for S such that ω(u, v) = 1. Then, u+ f(u), v + f(v) is a
basis for T . Moreover,
ι(S) · ι(T )vol = u ∧ v ∧ (u+ f(u)) ∧ (v + f(v)) = u ∧ v ∧ f(u) ∧ f(v).
We can compute which multiple of vol this last expression represents by using the
normalization (ω ∧ ω)(vol) = −2 and the computation
(ω ∧ ω)(u ∧ v ∧ f(u) ∧ f(v)) = 2Det(f).
We deduce that
ι(S) · ι(T ) = −Det(f).
40
Now we compute µ(S) · µ(T ) :
µ(S) · µ(T ) =
(
ι(S) +
1
2
ω(ι(S))ω∗
)
·
(
ι(T ) +
1
2
ω(ι(T ))ω∗
)
= −Det(f) + (1 + Det(f))− 1
2
(1 + Det(f))
= 1/2(1− Det(f)).
Finally, by the proof of Lemma 3.2.3, µ(S) · µ(S) = 1
2
and µ(T ) · µ(T ) = 1
2
(1 +
Det(f))2. Combining these computations finishes the proof of the statement.
3.3 Disjoint crooked surfaces
In this section we apply the techniques developed above in order to prove a
full disjointness criterion for pairs of crooked surfaces in Ein.
We work in the Lagrangian framework of Section 2.2.3 with the symplectic
vector space (V, ω).
Let u+, u−, v+, v− be four vectors in V such that
ω(u+, v−) = ω(u−, v+) = 1
and all other products between these four vanish. This means that we have La-
grangians
P0 := Rv+ + Rv−,
P∞ := Ru+ + Ru−, and
P± := Rv± + Ru±
representing the points of intersection of the photons associated to [u+], [u−], [v+], [v−].
We call this configuration of four points and four photons a lightlike quadrilateral.
41
The crooked surface C determined by this configuration is a subset of Ein
consisting of three pieces : two wings and a stem. The two wings are foliated by
photons, and we will denote by W+,W− the sets of photons covering the wings.
Each wing is a subset of the light cone of P+ and P−, respectively. Identifying
points in P(V ) with the photons they represent, the foliations are as follows:
W+ = {[tu+ + sv+] | ts ≥ 0},
W− = {[tu− + sv−] | ts ≤ 0}.
We will sometimes abuse notation and use the symbol W± to denote the collection
of points in the Einstein universe which is the union of these collections of photons.
The stem S is the subset of the Einstein torus determined by the splitting
S1 ⊕ S2 := (Ru+ + Rv−) ⊕ (Ru− + Rv+) consisting of timelike points with respect
to P0, P∞ :
S = {Rw + Rw′ | w ∈ S1, w′ ∈ S2, |m(P0, L, P∞)| = 2}.
Note that this definition gives only the interior of the stem as defined in [CFLD14].
This crooked surface is the closure in Ein of a crooked plane in the Minkowski patch
defined by the complement of the light cone of P∞.
Theorem 3.3.1. Let C1, C2 be two crooked surfaces with intersecting stems. Then,
the stem of C1 intersects a wing of C2 or vice versa. That is, crooked surfaces cannot
intersect in their stems only.
Proof. The stem consists of two disjoint, contractible pieces. To see this, note that
this set is contained in the Minkowski patch defined by P∞. There, the Einstein
42
torus containing the stem is a timelike plane through the origin, and the timelike
points in this plane form two disjoint quadrants. Let K be the intersection of the
two Einstein tori containing the stems of C1 and C2. Then, K is noncontractible
in either tori (Corollary 3.1.1), so it can’t be contained in the interior of the stem.
Therefore, ` must intersect the boundary of the stem which is part of the wings.
Lemma 3.3.2. Let p0, p∞, p ∈ Ein be three points in the Einstein universe. The
point p is timelike with respect to p0, p∞ if and only if the intersection of the three
light cones of p, p0, p∞ is empty.
Proof. We work in the model of Ein given by lightlike lines in a vector space of
signature (3, 2). If p is timelike with respect to p0, p∞, then it lies on a timelike
curve which means that the subspace generated by p, p0, p∞ has signature (1, 2).
Therefore, its orthogonal complement is positive-definite and contains no lightlike
vectors, so the intersection of the light cones is empty. The converse is similar.
Lemma 3.3.3. A photon represented by a vector p ∈ V is disjoint from the crooked
surface C if and only if the following two inequalities are satisfied:
ω(p, v+)ω(p, u+) > 0
ω(p, v−)ω(p, u−) < 0.
Proof. Write p in the basis u+, u−, v+, v− :
p = au+ + bu− + cv+ + dv−.
43
Then,
a = ω(p, v−) b = ω(p, v+)
c = −ω(p, u−) d = −ω(p, u+).
The photon p is disjoint from W+ if and only if the following equation has no
solutions:
ω(p, tu+ + sv+) = 0.
This happens exactly when bd < 0. Similarly, p is disjoint from W− if and only
if ac > 0. These two equations are equivalent to the ones in the statement of the
Lemma, therefore it remains only to show that under these conditions, p is disjoint
from the stem.
The Lagrangian plane P representing the intersection of p with the Einstein
torus containing the stem is generated by p and au+ + dv−. We want to show that
P cannot intersect the stem in a point which is timelike with respect to P0, P∞.
The intersection of the light cones of P0 and P∞ consists of planes of the form:
R(su+ + tu−) + R(s′v+ + t′v−) where st′ + ts′ = 0. We want to show that no point
represented by such a plane is incident to P . Two Lagrangian planes are incident
when their intersection is a non-zero subspace. Equivalently, they are incident if
they do not span V . We have :
det(p, au+ + dv−, su+ + tu−, s′v+ + t′v−)
= (−bdss′ + catt′) det(u+, u−, v+, v−)
= k(bds2 + act2) det(u+, u−, v+, v−),
44
where t′ = kt, s′ = −ks, k 6= 0. There exist t, s making this determinant vanish
because bd, ac have different signs. This means that the point where p intersects
the Einstein torus containing the stem is not timelike and therefore outside the
stem.
Theorem 3.3.4. Two crooked surfaces C,C ′ given respectively by the configura-
tions u+, u−, v+, v− and u′+, u
′
−, v
′
+, v
′
− are disjoint if and only if the four photons
u′+, u
′
−, v
′
+, v
′
− do not intersect C and the four photons u+, u−, v+, v− do not inter-
sect C ′.
Proof. Let us first show that the wing W+ of C does not intersect C ′. By Lemma
3.3.3, it suffices to show that
ω(tu+ + sv+, v
′
+)ω(tu+ + sv+, u
′
+) > 0
and
ω(tu+ + sv+, v
′
−)ω(tu+ + sv+, u
′
−) < 0
for all s, t ∈ R such that st ≥ 0 (with s and t not both zero).
We have
ω(tu+ + sv+, v
′
+)ω(tu+ + sv+, u
′
+)
= t2ω(u+, v
′
+)ω(u+, u
′
+) + stω(u+, v
′
+)ω(v+, u
′
+)
+ stω(v+, v
′
+)ω(u+, u
′
+) + s
2ω(v+, v
′
+)ω(v+, u
′
+).
By hypothesis, neither u+, v+ intersect C
′, and neither u′+, v
′
+ intersect C. Therefore,
using again Lemma 3.3.3 and st ≥ 0, we see that each term in this sum is non-
45
negative and that at least one of them must be strictly positive. Therefore,
ω(tu+ + sv+, v
′
+)ω(tu+ + sv+, u
′
+) > 0.
The proof that
ω(tu+ + sv+, v
′
−)ω(tu+ + sv+, u
′
−) < 0
is similar. Therefore, W+ does not intersect C ′.
In an analogous way, one can show that W− does not intersect C ′. Therefore,
the wings of the crooked surface C do not intersect C ′. Hence, to show that C and
C ′ are disjoint, it only remains to show that the stem of C does not intersect C ′.
By symmetry, the wings of C ′ do not intersect C, which means in particular
that they do not intersect the stem of C. Consequently, the stem of C can only
intersect the stem of C ′. However, according to Theorem 3.3.1, if the stem of C
intersects the stem of C ′, it must necessarily intersect its wings as well, which is not
the case here. Therefore, we conclude that C and C ′ must be disjoint.
By Lemma 3.3.3, this disjointness criterion can be expressed explicitly as 16
inequalities (two for each of the 8 photons defining the two crooked surfaces). There
is some redundancy in these inequalities, but there does not seem to be a natural
way to reduce the system.
3.4 Anti de Sitter crooked planes
In this section, we show that the criterion for disjointness of anti de Sitter
crooked planes described in [DGK14] is a special case of Theorem 3.3.4, when em-
bedding the double cover of anti de Sitter space in the Einstein universe.
46
Theorem 3.4.1 ( [DGK14], Theorem 3.2). Let `, `′ be geodesic lines of H2 and
g ∈ PSL(2,R). Then, the AdS crooked planes defined by (I, `) and (g, `′) are disjoint
if and only if for any endpoints ξ of ` and ξ′ of `′, we have ξ 6= ξ′ and d(ξ, gξ′) −
d(ξ, ξ′) < 0.
In this criterion, the difference d(p, gq) − d(p, q) for p, q ∈ ∂H2 is defined as
follows : choose sufficiently small horocycles C,D through p, q respectively. Then,
d(p, gq) − d(p, q) := d(C,GD) − d(C,D) and this quantity is independent of the
choice of horocycles.
3.4.1 AdS as a subspace of Ein
Let V0 be a real two dimensional symplectic vector space with symplectic form
ω0. Denote by V the four dimensional symplectic vector space V = V0⊕V0 equipped
with the symplectic form ω = ω0 ⊕ −ω0. This vector space V will have the same
role as in Section 3.2.
The Lie group Sp(V0) = SL(V0) is a model for the double cover of anti de
Sitter 3-space. We will show how to embed this naturally inside the Lagrangian
Grassmannian model of the Einstein Universe in three dimensions.
Define
i : SL(V0)→ Gr(2, V )
f 7→ graph(f)
The graph of f ∈ Sp(V0) is a Lagrangian subspace of V = V0⊕ V0. This means that
i(SL(V0)) ⊂ Lag(V ) ∼= Ein. This map is equivariant with respect to the homomor-
47
phism:
SL(V0)× SL(V0)→ Sp(V )
(A,B) 7→ B ⊕ A.
The involution of Ein induced by the linear map
I ⊕−I : V0 ⊕ V0 7→ V0 ⊕ V0,
where I denotes the identity map on V0, preserves the image of i. It corresponds
to the two-fold covering SL(V0) → PSL(V0). The fixed points of this involution are
exactly the complement of the image of i, corresponding to the conformal boundary
of AdS.
3.4.2 Crooked surfaces and AdS crooked planes
As in [Gol15], we say that a crooked surface is adapted to an AdS patch if it is
invariant under the involution I ⊕−I. More precisely, two of the opposite vertices
are fixed (they lie on the boundary of AdS) and the two others are swapped. If
we denote the four photons by u−, u+, v−, v+, this means v− = (I ⊕ −I)u− and
v+ = (I ⊕−I)u+.
3.4.2.1 AdS crooked planes based at the identity
For concreteness, choose a basis of V to identify it with R4. We will represent a
plane in R4 by a 4×2 matrix whose columns generate the plane, up to multiplication
on the right by an invertible 2 × 2 matrix. For example, graph(f) corresponds to
48
the matrix: I
f
 .
The identity element of SL(V0) maps to the planeI
I

and its image under the involution I ⊕−I is I
−I
 .
In order to complete this to a lightlike quadrilateral, we choose a pair of vectors
a, b ∈ V0 (2×1 column vectors). Then, the four vertices of the lightlike quadrilateral
are: I
I
 ,
a a
a −a
 ,
b b
b −b
 ,
 I
−I
 .
We will say that such a lightlike quadrilateral is based at I and defined by the
vectors a, b. Its lightlike edges are the photons represented by vectors:
u+ =
a
a
 , u− =
−a
a

v+ =
b
b
 , v− =
 b
−b
 .
3.4.2.2 AdS crooked planes based at f
In order to get an AdS crooked plane based at a different point f ∈ SL(V0), we
map the crooked plane by an element of the isometry group SL(V0)×SL(V0) ⊂ Sp(V ).
49
The easiest way is to use an element of the form :I 0
0 f
 .
This corresponds to left multiplication by f in SL(V ).
Applying f to a lightlike quadrilateral, we get a lightlike quadrilateral with
vertices of the form: I
f
 ,
 I
−f
 ,
 a −a
fa fa
 ,
 b b
fb −fb

and edges of the form:  a
fa
 ,
−a
fa

 b
fb
 ,
−b
fb
 .
3.4.3 Disjointness
The disjointness criterion for crooked surfaces in the Einstein Universe is given
by 16 inequalities. Using the symmetries imposed by an AdS patch, we can reduce
them to 4 inequalities.
Using the involution defining the AdS patch, we can immediately reduce the
number of inequalities by half. This is because both surfaces are preserved by the
involution, and their defining photons are swapped in pairs. (So for example, we
only have to check that u+ and u− are disjoint from the other surface, for each
surface.)
50
The second reduction comes from the fact that for AdS crooked planes, we
only need to check that the four photons from the first crooked surface are disjoint
from the second, and then the four from the second are automatically disjoint from
the first.
For a crooked surface based at the identity with lightlike quadrilateral defined
by the vectors a, b ∈ V0 and another based at f with quadrilateral defined by
a′, b′ ∈ V0, the inequalities reduce to:
ω0(a
′, b)2 > ω0(fa′, b)2
ω0(a
′, a)2 > ω0(fa′, a)2
ω0(b
′, b)2 > ω0(fb′, b)2
ω0(b
′, a)2 > ω0(fb′, a)2.
What remains is to interpret these four inequalities in terms of hyperbolic geometry.
We first define an equivariant map from P(V0) to ∂H2. As a model of the boundary
of H2, we use the projectivized null cone for the Killing form in sl(2,R). Define
η : V0 → N(sl(2,R))
a 7→ −aaTJ,
where a is a column vector representing a point in P(V0). This map associates to
the vector a the tangent vector to the identity of the photon between I and the
boundary point
a a
a −a
. Note that the image of η is contained in the upper part
of the null cone.
51
Lemma 3.4.2. η is equivariant with respect to the action of SL(V0).
Proof.
η(Aa) = −Aa(Aa)TJ = −AaaTATJ = −AaaTJA−1 = Aη(a)A−1.
Lemma 3.4.3. Let a, b ∈ V0. Then, ω0(a, b)2 = −K(η(a), η(b)).
Proof.
ω0(a, b)
2 = −aTJbbTJa
= aTJη(b)a
= Tr(aTJη(b)a
= Tr(aaTJη(b))
= −Tr(η(a)η(b))
= −K(η(a), η(b)).
Note that the expression ω0(a, b) is not projectively invariant, but the sign of
ω0(a, b)
2 − ω0(a, fb)2 is.
Corollary 3.4.4. The following inequalities are equivalent
ω0(a, b)
2 − ω0(a, fb)2 > 0,
K(η(a), fη(b)f−1) > K(η(a), η(b)).
52
Finally, we want to show that the four inequalities above imply the DGK cri-
terion. Let A,B,A′, B′ denote respectively η(a), η(b), η(a′), η(b′). Then, A,B,A′, B′
represent endpoints of two geodesics g, g′ in the hyperbolic plane. We want to show
d(ξ, fξ′f−1)− d(ξ, ξ′) < 0
for ξ ∈ {A,B} and ξ′ ∈ {A′, B′}.
We use the hyperboloid model of H2, {X ∈ sl(2,R) | K(X,X) = −1}.
Consider horocycles Cξ(r) = {X ∈ H2 | K(X, ξ) = −r} and Cξ′(r′) = {X ∈
H2 | K(X, ξ′) = −r′} at ξ and ξ′ respectively. The distance between these two
horocycles is given by the formula
d(Cξ(r), Cξ′(r
′)) = arccosh
(
−1
2
(
K(ξ, ξ′)
2rr′
+
2rr′
K(ξ, ξ′)
))
.
Similarly,
d(Cξ(r), fCξ′(r
′)f−1) = arccosh
(
−1
2
(
K(ξ, fξ′f−1)
2rr′
+
2rr′
K(ξ, fξ′f−1)
))
.
We know that K(ξ, fξ′f−1) > K(ξ, ξ′). If r, r′ are sufficiently small, by increasing-
ness of the function x 7→ x+ 1
x
for x > 1 and increasingness of arccosh we conclude
d(Cξ(r), Cξ′(r
′)) > d(Cξ(r), fCξ′(r′)), which is what we wanted.
53
Chapter 4: Partial cyclic orders
The projective line RP1 admits a cyclic order invariant under the action of
projective automorphisms. The simplest Schottky groups are defined by their action
on RP1 by such automorphisms. Indeed, the application of the Ping-pong Lemma
only relies on this cyclic order, and it is possible to define Schottky subgroups of
the orientation-preserving homeomorphism group Homeo+(RP1). This motivates
the idea of finding analogs of this cyclic order in other spaces in order to define
Schottky groups in a broader context. The contents of this chapter are mostly from
the joint preprint with N. Treib [BT16].
4.1 Definitions
A partial cyclic order is a relation on triples which is analogous to a partial
order, but generalizing a cyclic order instead of a linear order. The definition we
use was introduced in 1982 by Nova´k [Nov82].
Definition 4.1.1. A partial cyclic order (PCO) on a set C is a relation −→ on triples
in C satisfying, for any a, b, c, d ∈ C :
• if −→abc, then −→bca (cyclicity).
• if −→abc, then not −→cba (asymmetry).
54
• if −→abc and −→acd, then −→abd (transitivity).
If in addition the relation satisfies:
• If a, b, c are distinct, then either −→abc or −→cba (totality),
then it is called a total cyclic order.
Let C,D be partially cyclically ordered sets.
Definition 4.1.2. A map f : C → D is called increasing if −→abc implies −−−−−−−−→f(a)f(b)f(c).
An automorphism of a partial cyclic order is an increasing map f : C → C with an
increasing inverse. We will denote by G the group of all automorphisms of C.
Any subset X ⊂ C such that the restriction of the partial cyclic order is a
total cyclic order on X will be called a cycle. We will also use the term cycle for
(ordered) tuples (x1, . . . , xn) ∈ Cn if the cyclic order relations between the points in
C agree with the cyclic order given by the ordering of the tuple.
Definition 4.1.3. Let a, b ∈ C. The interval between a and b is the set (a, b) :=
{x ∈ C | −→axb}. The set of all intervals generates a natural topology on C under
which automorphisms of the partial cyclic order are homeomorphisms. We call this
topology the interval topology on C. We call C first countable when its interval
topology is first countable. We need this last condition to justify the use of the
sequential definition of continuity, for instance in the proof of Theorem 4.2.7.
The opposite of an interval I = (a, b) is the interval (b, a), also denoted by −I.
Example 4.1.4. The circle S1 admits a total cyclic order. The relation on triples
is :
−→
abc whenever (a, b, c) are in counterclockwise order around the circle. The au-
55
tomorphism group of this cyclic order is the group of orientation preserving home-
omorphisms of the circle.
Example 4.1.5. We can define a product cyclic order on the torus S1×S1. Define
the relation to be −−→xyz whenever −−−−→x1y1z1 and −−−−→x2y2z2. This is not a total cyclic order.
Some intervals in this cyclically ordered space are shown in Figure 4.2.
Example 4.1.6. Every strict partial order < on a set X induces a partial cyclic
order in the following way: define
−→
abc if and only if either a < b < c, b < c < a, or
c < a < b. The cyclic permutation axiom is automatic and the two other axioms
follow from the antisymmetry and transitivity axioms of a partial order.
The key topological property that we will need in the next section is a notion
of completeness that we can associate to a space carrying a PCO.
Definition 4.1.7. A sequence a1, a2, · · · ∈ C is increasing if and only if −−−→aiajak
whenever i < j < k.
Equivalently, the map a : N→ C defined by a(i) = ai is increasing, where the
cyclic order on N is given by
−→
ijk whenever i < j < k, j < k < i or k < i < j (as in
Example 4.1.6).
Definition 4.1.8. A partially cyclically ordered set C is increasing-complete if every
increasing sequence converges to a unique limit in the interval topology.
The following is a natural equivalence relation for increasing sequences.
Definition 4.1.9. Two increasing sequences an and bm are called compatible if they
admit subsequences ank and bml making the combined sequence an1 , bm1 , an2 , bm2 , . . .
increasing.
56
Lemma 4.1.10. Let C be an increasing-complete partially cyclically ordered set,
and let an and bm be compatible increasing sequences. Then their limits agree.
Proof. Any increasing sequence has a unique limit, and any subsequence of an in-
creasing sequence therefore has the same unique limit.
The combined sequence (see the previous definition) is increasing, hence its unique
limit must agree with the unique limits of both subsequences ank and bml .
To complete this list of definitions related to PCOs, we finish with two further
restrictions on a set with a PCO which will be useful in Section 4.2.2.
Definition 4.1.11. A partially cyclically ordered set C is proper if for any increasing
quadruple (a, b, c, d) ∈ C4, we have (b, c) ⊂ (a, d). Here, “bar” denotes the closure
in the interval topology.
Definition 4.1.12. Two points a, b ∈ C in a partially cyclically ordered set C are
called comparable if there exists a point c ∈ C with either −→abc or −→acb.
Definition 4.1.13. A PCO set C is full if whenever (a, b) is non-empty for some
pair a, b, then (b, a) is also non-empty. Equivalently, whenever a, b are comparable
then both intervals they bound are non-empty.
Remark 4.1.14. The motivation for the term “full” stems from the following con-
struction. Assume we have a non-empty interval (a, b). Then we can find a point
c ∈ (a, b) and another point d ∈ (b, a). But then the point d also lies in the interval
(c, a), so by fullness, there is a point inside (a, c) as well. Continuing in this fashion,
we can subdivide all resulting intervals further and further, and thereby construct a
57
countably infinite subset X ⊂ (a, b) with the following two properties: Firstly, X
is a cycle. Secondly, for any pair x1, x2 of distinct elements of X, the intersection
(x1, x2) ∩X is nonempty.
4.2 Generalized Schottky groups
Throughout this section, C denotes a partially cyclically ordered set and G =
Aut(C).
4.2.1 Definition of generalized Schottky group
Let Σ be the interior of a compact, connected, oriented surface with boundary
of Euler characteristic χ < 0. Then, the fundamental group pi1(Σ) is free on g = 1−χ
generators. Let Γ ⊂ PSL(2,R) be the holonomy of a finite area hyperbolization of
Σ. In this section, we construct free subgroups of G using Γ as a combinatorial
model.
It is well known that there is a presentation for Γ of the following form : Γ is
freely generated by A1, . . . , Ag ∈ PSL(2,R) and there are 2g disjoint open intervals
I+1 , . . . , I
+
g , I
−
1 , . . . , I
−
g ⊂ RP1 ∼= S1 such that Aj(−I−j ) = I+j . Moreover, we have
that
⋃
i
I+i ∪
⋃
i
I−i = RP
1 (Figure 4.1).
The cyclic ordering on S1 gives a cyclic ordering to the intervals in the defini-
tion.
We call a k-th order interval the image of any I+j (respectively I
−
j ) by a reduced
word W = γ1γ2 . . . γk−1 of length k − 1 with γk−1 6= A−1j (respectively γk−1 6= Aj).
58
Figure 4.1: A combinatorial model for the once punctured torus.
There are exactly (2g)(2g − 1)k−1 k-th order intervals. There is a natural bijection
between words of length k and k-th order intervals. We use this bijection to index
k-th order intervals : IW is the interval corresponding to the word W . For any
fixed k, the k-th order intervals are all pairwise disjoint, and so they are cyclically
ordered. This induces a cyclic ordering on words of length k in Γ. The union of all
closures of k-th order intervals is all of RP1.
The following easy lemma, which is a reformulation of transitivity, motivates
our definition of generalized Schottky groups in G.
Lemma 4.2.1. Let (a, b, c) ∈ C3 be a cycle. Then we have (b, c) ⊂ (b, a). In
particular, the intervals (a, b) and (b, c) are disjoint.
Proof. Let x ∈ (b, c), so we have −→bxc. By transitivity, together with −→bca, this implies
59
−→
bxa.
We now define generalized Schottky subgroups of G by asking for a setup of
intervals similar to the PSL(2,R) case and requiring generators to pair them the
same way.
Definition 4.2.2. Let ξ0 be an increasing map from the set of endpoints of the
intervals I+1 , . . . , I
+
g , I
−
1 , . . . , I
−
g into C. For I
±
i = (a
±
i , b
±
i ), define the corresponding
interval J±i = (ξ0(a
±
i ), ξ0(b
±
i )) ⊂ C. Assume there exist h1, . . . , hg ∈ G which pair
the endpoints of J±i in the same way that the gi pair the I
±
i , so that hi(−J−i ) =
J+i . We call the image of the induced homomorphism Γ → G sending Ai to hi a
generalized Schottky group, and the intervals J±i used to define it a set of Schottky
intervals for this group.
Remark 4.2.3.
1. A generalized Schottky group will in general have many possible choices of a
set of Schottky intervals. We will only use this term when a specific choice of
both generators and intervals is fixed.
2. Since the cyclic ordering is a property of RP1 which is not shared by CP1,
the Schottky groups defined here do not generalize the more well known CP1
Kleinian case.
3. The requirement that the combinatorial model be a finite-area hyperbolization
is artificial. It is helpful in order to avoid having to separate our analysis into
several cases. We could use a model where the intervals have disjoint closures
and the construction would work in the same way. Such models always admit
60
a choice of Schottky generators with contiguous Schottky intervals as well, so
they form a strict subset. In Section 4.4.2 we will use intervals with disjoint
closures to describe domains of discontinuity in RP2n−1.
4. Our use of the term “Schottky” differs slightly from most references in that we
allow for the closures of the Ping-pong subsets to intersect. This is sometimes
called “Schottky-type”.
With this setup, we can define k-th order intervals in C in the same way as
above but starting with the intervals J±i and their images under words in the hi (see
Figure 4.2). As above, denote by JW the interval corresponding to W . Note that
since ξ0 is increasing, the k-th order intervals in C are also cyclically ordered, where
the ordering is the same as the ordering of the corresponding intervals in RP1.
Proposition 4.2.4. The group generated by h1 . . . hg is free on those generators.
Proof. Define Ji = J
+
i ∪J−i . Note that Ji∩Jj = ∅ whenever i 6= j. Moreover, for any
n 6= 0, hni (Jj) ⊂ Ji and so the proposition follows from the Ping-pong lemma.
The endpoints of k-th order intervals in C satisfy the same cyclic order rela-
tions as the corresponding endpoints in S1, and we can extend ξ0 to an increasing
equivariant map defined on the countable dense set of all endpoints of k-th order
intervals in S1. We denote this set by S1Γ.
4.2.2 Limit curves
Lemma 4.2.5. Let C be a partially cyclically ordered set which is full and proper,
and (x1, . . . , x6) ∈ C6 a cycle. Let I1 = (x1, x2), I2 = (x3, x4), I3 = (x5, x6) and
61
Figure 4.2: Some first, second and third order intervals for a generalized Schottky
group acting on S1 × S1.
62
ai ∈ Ii be arbitrary points in the closures of the intervals. Then −−−−→a1a2a3.
Proof. Since C is full, we can choose auxiliary points yi such that the 12-tuple
(y1, x1, x2, y2, y3, x3, x4, y4, y5, x5, x6, y6) is a cycle. This allows us to conclude that
(xi, xi+1) ⊂ (yi, yi+1) for odd i as C is proper. Since (y1, . . . , y6) is a cycle, transitivity
implies the lemma.
Lemma 4.2.6. Let Pn → P be an increasing sequence in a proper, increasing
complete, PCO set C. Assume Qn is another sequence with Qn ∈ (Pn, Pn+1) for
all n. Then Qn converges to P and is 3-increasing in the following sense: whenever
i+ 2 < j < k − 2, we have −−−−−→QiQjQk.
Proof. For every n ≥ 2, Qn ∈ (Pn−1, Pn+2) by properness, which already implies
that Qn is 3-increasing. Now, consider the following sequence:
P1, Q2, P4, Q5, . . . , P3n+1, Q3n+2, . . .
It is increasing, and admits a subsequence which is also a subsequence of Pn. Since
increasing sequences have unique limits, this sequence must converge to P . The
increasing subsequence Q3n+2 therefore converges to P . Using the same argument,
we see that Q3n+1 and Q3n also converge to P , so in fact the sequence Qn converges
to P .
We now come to the main theorem of this section, which explains how to
construct a boundary map for generalized Schottky groups, under some topological
assumptions.
The boundary map we construct will be left-continuous as a map from S1 to some
63
first countable topological space C. To avoid confusion, let us fix the definition here:
In a small neighborhood U of a point a ∈ S1, the cyclic order induces a linear order.
A sequence an ∈ U converges to a from the left if an < a and an n→∞−−−→ a. The
function f is left-continuous at a if f(an)→ f(a) for all sequences an converging to
a from the left.
It is worth noting that to check for left-continuity at a point a, it is in fact sufficient
to check the convergence of f(an) for increasing sequences an converging to a from
the left. The reason is the following: Assume an is a sequence converging to a from
the left such that f(an) does not converge to f(a). Then it has a subsequence such
that f(ank) stays bounded away from f(a). But since ank → a from the left, we
can pick a further subsequence which is increasing and still a counterexample to
left-continuity.
Theorem 4.2.7. Let ρ : Γ → G be the map defining a generalized Schottky group.
Assume that C is first countable, increasing-complete, full and proper. Then there
is a left-continuous, equivariant, increasing boundary map ξ : S1 → C.
Proof. We construct the map ξ as follows: Recall that S1Γ ⊂ S1 denotes the domain
of ξ0 and is a dense subset. For x ∈ S1, pick any increasing sequence xn ∈ S1Γ
converging to x and set
ξ(x) = lim
n→∞
ξ0(xn).
First of all, let us show that this value is well-defined. Since xn is an increasing
sequence, the increasing map ξ0 maps it to an increasing sequence in C which
therefore has a unique limit. Furthermore, this limit does not depend on the choice of
64
xn: Let ym be another increasing sequence converging to x. Then the two sequences
ξ0(xn) and ξ0(ym) are compatible, so they have the same limit by Lemma 4.1.10.
We then verify that ξ is equivariant. Let x ∈ S1, γ ∈ Γ, and xn → x an
increasing sequence, so we have ξ(x) = lim
n→∞
ξ0(xn). Then γ(xn) is an increasing
sequence converging to γ(x), so by continuity of ρ(γ) and equivariance of ξ0, we
have the following equalities:
ρ(γ)(ξ(x)) = lim
n→∞
ρ(γ)(ξ0(xn)) = lim
n→∞
ξ0(γ(xn)) = ξ(γ(x)).
Next, we show that it is left-continuous. Assume xn ∈ S1 is a sequence
converging to x from the left. As explained above, without loss of generality we
can take xn to be an increasing sequence. We pick points yn ∈ S1Γ such that yn ∈
(xn−1, xn). Then yn is increasing and xn ∈ (yn, yn+1). Furthermore, yn also converges
to x, hence
ξ(x) = lim ξ0(yn). (4.1)
Now, for each n, let {ak(n)}k∈N ⊂ S1Γ be an increasing sequence converging to xn,
so
ξ(xn) = lim
k→∞
ξ0(ak(n)). (4.2)
Then ak(n) ∈ (yn, yn+1) for large k, so
lim
k→∞
ξ0(ak(n)) ∈ (ξ0(yn), ξ0(yn+1)) (4.3)
because ξ0 is increasing. Now Lemma 4.2.6 applies and, combined with (4.1), (4.2)
and (4.3), tells us that ξ(xn) converges to ξ(x).
The final property we need to check is that ξ is increasing. Assume that we
have−−→xyz for points x, y, z ∈ S1. By density of S1Γ, we can find a cycle (a1, a2, b1, b2, c1, c2) ∈
65
(S1Γ)
6 such that x ∈ (a1, a2), y ∈ (b1, b2), z ∈ (c1, c2). As in the proof of left-
continuity, this implies that ξ(x) ∈ (ξ0(a1), ξ0(a2)), and similar for the other two
points. Using Lemma 4.2.5, we conclude
−−−−−−−−→
ξ(x)ξ(y)ξ(z).
The very general construction described in this section applies to many exam-
ples. For instance, various notions of positivity in homogeneous spaces give rise to
partial cyclic orders. More specifically, the Shilov boundary of Hermitian symmetric
spaces admits a PCO satisfying all the above properties, and the next section dedi-
cated to this example. It is also possible, using techniques similar to Fock-Goncharov
total positivity [FG06], to construct a PCO on spaces of complete oriented flags.
We will explain this partial cyclic order in Section 4.5 and how it can be used to
describe convex projective structures on surfaces with boundary (see also [BT17]).
4.3 Hermitian symmetric spaces of tube type
In this section, we show that the Shilov boundary of a Hermitian symmetric
space of tube type X admits a partial cyclic order invariant under the biholomor-
phism group of X
A motivating example is the case where X is the Siegel upper half space of
2× 2 complex matrices with positive-definite imaginary part. The Shilov boundary,
in this case, identifies with the Einstein universe. The partial cyclic order arises
from the causal structure on Ein.
We prove that Shilov boundaries satisfy the topological assumptions from The-
orem 4.2.7, so we have a boundary map for every generalized Schottky subgroup.
66
Then, using the machinery of Section 4.2, we show that Schottky subgroups in this
case correspond to maximal representations.
4.3.1 The partial cyclic order on the Shilov boundary
Let V be a real Euclidean vector space. That is, V is equipped with a scalar
product 〈, 〉.
Definition 4.3.1. A symmetric cone Ω ⊂ V is an open convex cone which is self-
dual and homogeneous. More precisely, the dual cone
Ω∗ := {v ∈ V | 〈u, v〉 > 0,∀u ∈ Ω\{0}}
equals Ω itself, and the subgroup of GL(V ) preserving Ω acts transitively on Ω.
A tube type domain is a domain of the form X = V + iΩ ⊂ VC in the complexi-
fication of V , where Ω is a symmetric cone. Let G be the group of biholomorphisms
of X.
The vector space V admits a Euclidean Jordan algebra structure associated
to the symmetric cone Ω. The two structures (symmetric cone and Jordan algebra)
determine each other [FK94].
Definition 4.3.2. A Jordan algebra is a vector space V over R together with a
bilinear product (u, v) 7→ uv ∈ V satifsying:
uv = vu
and
u(u2v) = u2(uv)
67
for all u, v ∈ V .
Definition 4.3.3. A Jordan algebra V is Euclidean if it admits an identity element
e, and there exists a positive definite inner product 〈, 〉 on V such that
〈uv, w〉 = 〈v, uw〉
for all u, v, w ∈ V . The cone of squares of V is
C = {v2 | v ∈ V }.
The interior C◦ of C is a symmetric cone, and coincides with Ω for the Jordan
algebra structure induced by Ω.
Example 4.3.4. Consider V = R2,1 a 3-dimensional real vector space with Lorentzian
inner product u · v = u1v1 + u2v2 − u3v3. The set Ω = {v ∈ V | v · v < 0, v3 > 0} of
future-pointing timelike vectors is a symmetric cone. The Jordan algebra structure
associated to this cone is given by the product:
(u1, u2, u3)(v1, v2, v3) = (u1v3 − u3v1, u2v3 − u3v2, u1v1 + u2v2 + u3v3).
Example 4.3.5. The set of n×n real symmetric matrices is a Jordan algebra with
product A ? B = (AB + BA)/2. The corresponding symmetric cone is the cone of
positive-definite matrices.
There is a spectral theorem for Euclidean Jordan algebras :
Proposition 4.3.6 ( [FK94] Theorem III.1.2). Let v ∈ V with dim(V ) = k. Then,
there exist unique real numbers λ1, . . . , λk, and a Jordan frame of primitive orthog-
onal idempotents c1, . . . , ck (that is, c
2
i = ci, cicj = 0 for i 6= j, and
∑
ci = e) such
68
that
v = λ1c1 + . . . λkck.
The λi are called the eigenvalues of v.
Definition 4.3.7. The partial order <Ω on a Jordan algebra V is defined by x <Ω y
if and only if y − x ∈ Ω.
The Cayley transform is the classical biholomorphic map which sends the
upper half plane to the unit disk in C. We will use the following generalization to
Jordan algebras in order to define a bounded realization of tube type domains.
Definition 4.3.8. Let D = {z ∈ VC | z+ ie is invertible}, where e is the identity of
the Jordan algebra and we extend the multiplication linearly to the complexification
of V .
The Cayley transform is the map p : D→ VC defined by
p(v) = (v − ie)(v + ie)−1.
Proposition 4.3.9 ( [FK94], Theorem X.4.3). The Cayley transform p maps the
tube type domain X = V ⊕ iΩ biholomorphically onto a bounded domain B ⊂ VC,
which we call the bounded domain realization of X (also known as the Harish-
Chandra realization).
Definition 4.3.10. If B is a bounded domain in Cn, denote by C(B) the set of
continuous functions on B which are holomorphic on B. The Shilov boundary S of
B is the smallest closed subset of ∂B such that, for all f ∈ C(B) we have
max
z∈B¯
|f(z)| = max
z∈S
|f(z)|.
69
By extension, the Shilov boundary of a tube type domain X is the Shilov boundary
of its bounded domain realization. The action of the group G of biholomorphisms
of B extends smoothly to its Shilov boundary.
Proposition 4.3.11 ( [FK94], Proposition X.2.3). The Cayley transform p : V →
VC maps the vector space V into the Shilov boundary S and p(V ) = S.
Using the following notion of transversality, we can make the previous propo-
sition more precise and say explicitly which points are in the image of p.
Definition 4.3.12. Two points x, y ∈ S are called transverse if the pair (x, y) ∈
S × S belongs to the unique open G-orbit for the diagonal action.
The image of the Cayley transform is exactly the set of points x ∈ S which
are transverse to a fixed point which we denote by ∞. [Wie04, Section 6.6.1]
The next object we need to define is the generalized Maslov index (generalizing
the case of the Lagrangian Grassmannian in Definition 2.2.14). This index is a
function on ordered triples of points in S, invariant under G. It will be used in order
to define a partial cyclic order on S, extending the partial cyclic order induced by
<Ω on p(V ) ⊂ S.
The generalized Maslov index is defined in [Cle04] using the notion of Γ-radial
convergence. For our purposes we will use the following equivalent definition, given
in the same paper.
Definition 4.3.13. Let x, y, z ∈ S. Applying an element of G, we may assume
x, y, z ∈ p(V ). Let vx, vy, vz ∈ V be the vectors which map respectively to x, y, z
70
under the Cayley transform p. Then, the generalized Maslov index of x, y, z is the
integer
M(x, y, z) := k(vy − vx) + k(vz − vy) + k(vx − vz)
where k(v) is the difference between the number of positive eigenvalues of v and the
number of negative eigenvalues of v in its spectral decomposition.
When x, y are transverse to z, equivalently, we can map z to ∞ using an
element of G and define
M(x, y,∞) = k(vy − vx)
Proposition 4.3.14. The Maslov index enjoys the following properties :
• G-invariance : M(gx, gy, gz) = M(x, y, z).
• Skew-symmetry : M(x1, x2, x3) = sgn(σ)M(xσ(1), xσ(2), xσ(3)). (for any permu-
tation σ ∈ S3)
• Cocycle identity : M(y, z, w)−M(x, z, w) + M(x, y, w)−M(x, y, z) = 0.
• Boundedness : |M(x, y, z)| ≤ rk(X)
These properties allow us to define a partial cyclic order on the Shilov bound-
ary.
Proposition 4.3.15. The relation −−→xyz if and only if M(x, y, z) = rk(X) defines a
G-invariant partial cyclic order on S.
Proof. Since M is skew-symmetric, the relation automatically satisfies the first two
axioms of a partial cyclic order. To prove the third axiom, assume M(x, y, z) =
71
M(x, z, w) = rk(X). By the cocycle identity,
M(y, z, w)−M(x, z, w) + M(x, y, w)−M(x, y, z) = 0
and so
M(y, z, w) + M(x, y, w) = 2 rk(X)
which is only possible if M(y, z, w) = M(x, y, w) = rk(X).
The Shilov partial cyclic order −→ is closely related with the causal structure
on S introduced by Kaneyuki [Kan91]. Namely, whenever −−→xyz, there is a future-
oriented closed timelike curve going through x, y, z in that order. Informally, y is
in the intersection of the future of x and the past of z. The following two lemmas
describe some immediate properties of cyclically ordered triples.
Lemma 4.3.16 ( [Wie04], Lemma 5.5.4). Let x, y, z ∈ S with −−→xyz. Then x, y, z are
pairwise transverse.
Lemma 4.3.17. Assume x, y ∈ V . Then, −−→xy∞ if and only if x <Ω y.
Proof. The cone Ω coincides with the region where k(v) = rk(X).
Remark 4.3.18. The interval topology on S is the same as the usual manifold
topology.
Proposition 4.3.19. The PCO defined by −→ on S is increasing-complete, full and
proper.
Proof. We first show that it is increasing-complete. Let x1, x2, . . . be an increasing
sequence in S. Let g ∈ G be such that gx2 = ∞. Then, since we have −−−−−−→xkxk+1x2
72
for all k ≥ 3, the sequence gx3, gx4, . . . is an increasing sequence transverse to ∞.
Hence, there exist v3, v4, . . . ∈ V with p(vk) = gxk.
This new sequence is increasing with respect to <Ω. Moreover, it is bounded
since we have −−−−−−−→gxkgx1gx2 for all k > 2, so vk <Ω v1 where p(v1) = gx1. The tail of the
sequence is contained in (v3, v1) which is compact, so it has an accumulation point.
If w,w′ are two accumulations points of the sequence, let wk, w′k be subsequences
converging respectively to each of them. Passing to subsequences if necessary, we
can arrange so that wk <Ω w
′
k for all k, and so w
′
k−wk ∈ Ω. This implies w′−w ∈ Ω,
and by the same argument we can also show w−w′ ∈ Ω. Since Ω is a proper convex
cone (in the sense of [FK94]), its closure does not contain any opposite pairs, so
w = w′.
Now we turn to fullness of the PCO. Whenever an interval (x, y) is nonempty,
its endpoints have to be transverse by Lemma 4.3.16. We can therefore apply an
element of G to map x to∞ and y inside p(V ). Then Lemma 4.3.17 shows that the
interval (y, x) is also nonempty.
Finally, we show that the PCO is proper. Let (x1, x2, x3, x4) ∈ S4 be a cycle.
Using an element of G, we can assume that x4 is ∞, so that x1, x2, x3 ∈ p(V ). Let
vi ∈ V be the vector such that p(vi) = xi for i = 1, 2, 3. Now the cyclic relations
−−−−→x1x2∞ and −−−−→x2x3∞ imply that both v2−v1 and v3−v2 lie in the cone Ω. The interval
(x2, x3) is therefore given by p ((v2 + Ω) ∩ (v3 − Ω)). This implies the claim since
(v2 + Ω) ∩ (v3 − Ω) is a relatively compact set in V whose closure is contained in
v1 + Ω, which is mapped onto (x1,∞) by p.
73
4.3.2 Maximal representations
In the previous section we defined a PCO on the Shilov boundary S of a Her-
mitian symmetric space of tube type on which the group of holomorphic isometries
G acts by order-preserving diffeomorphisms. We recall that this action is transitive
on transverse pairs. The Schottky construction described in Section 4.2 therefore
gives maps ρ : Γ→ G where Γ is the fundamental group of a surface with boundary.
Maximal representations are a class of geometrically interesting representations and
we will show in this section that they correspond to Schottky subgroups. They are
defined by associating a natural invariant to the representation and requiring it to
attain its maximal possible value. While the study of this invariant was originally
restricted to closed surfaces ([Tol79], [DT87], [Tol89]), the definition was extended
to surfaces with boundary in [BIW10].
Let X be a Hermitian symmetric space and ω be the Ka¨hler form on X.
Then, ω defines a continuous, bounded cohomology class κbG ∈ H2cb(G,R) called the
Ka¨hler class. If ρ : pi1(Σ) → G is a representation, the pullback ρ∗κbG is a bounded
cohomology class in H2b (pi1(Σ),R) ∼= H2b (Σ,R). In order to get an invariant out of
this class, we use the isomorphism j : H2b (Σ, ∂Σ,R) → H2b (Σ,R) (see [BIW10] for
details).
Definition 4.3.20. The Toledo invariant is the real number
T(ρ) = 〈j−1ρ∗κbG, [Σ, ∂Σ]〉
where [Σ, ∂Σ] is the relative fundamental class.
74
The Toledo invariant satisfies a sharp bound generalizing the Milnor-Wood-
inequality : |T(ρ)| ≤ |χ(Σ)| rk(X). A representation ρ is called maximal whenever
equality is attained. The key to our analysis is the following characterization from
[BIW10] (Theorem 8) :
Theorem 4.3.21. Let h : Γ → PSL(2,R) be a complete finite area hyperbolization
of the interior of Σ and ρ : Γ → G a representation into a group of Hermitian
type. Then ρ is maximal if and only if there exists a left continuous, equivariant,
increasing map
ξ : S1 → S
where S is the Shilov boundary of the bounded symmetric domain associated to G.
Using this characterization and our earlier construction of a boundary map for
generalized Schottky representations, we see that the two notions agree:
Theorem 4.3.22. The representation ρ : Γ→ G is maximal if and only if it admits
a Schottky presentation.
Proof. Assume ρ is Schottky. Proposition 4.3.19 states that all the prerequisites of
Theorem 4.2.7 are fulfilled. Therefore, there exists a boundary map ξ satisfying the
conditions of the characterization above, so ρ is maximal.
Conversely, if ρ is maximal, then we have such a map ξ. Choosing a Schottky
presentation for the hyperbolisation h, we get a Schottky presentation for ρ by using
the intervals (ξ(a), ξ(b)) where (a, b) is some Schottky interval in the presentation
for h. Equivariance and positivity of ξ ensure that these intervals fit our definition
of generalized Schottky groups.
75
Theorem 4.3.22, as stated, assumes that G is of tube type. However, this
assumption is not necessary. This is because of the following observations. Let X
be a Hermitian symmetric space, and S its Shilov boundary. Then, in the same way
as for tube type, the generalized Maslov index defines a partial cyclic order on S. Let
x, y ∈ S be transverse. Then, x, y are contained in the Shilov boundary of a unique
maximal tube type subdomain of X [Wie04, Lemma 4.4.2]. Moreover, this is also
true of any increasing triple in S [Wie04, Proposition 5.1.4]. This means that any
increasing subset of S is contained in the Shilov boundary of a tube type subdomain,
and so the proofs of this section generalize to arbitrary Hermitian symmetric spaces.
4.4 Schottky groups in Sp(2n,R)
In this section, we consider the symplectic group Sp(2n,R), acting on R2n
equipped with a symplectic form ω, and describe the construction of Schottky groups
in detail.
4.4.1 The Maslov index in Sp(2n,R)
Definition 4.4.1. Let P,Q be transverse Lagrangians in R2n. We associate to them
an antisymplectic involution σPQ defined using the splitting R2n = P ⊕Q:
σPQ : P ⊕Q→ P ⊕Q
(v, w) 7→ (−v, w)
We call this antisymplectic involution the reflection in the pair P,Q. This generalizes
the projective reflection in RP1.
76
We will sometimes abuse notation and use σPQ to denote the induced trans-
formation on Grassmannians.
Using this involution, we associate a symmetric bilinear form to the pair P,Q
:
Definition 4.4.2.
BPQ(v, w) := ω(v, σPQ(w))
This bilinear form is nondegenerate and has signature (n, n).
Definition 4.4.3. Let P,Q,R be pairwise transverse Lagrangians in R2n. The
Maslov index of the triple (P,Q,R) is the index of the restriction of BPR to Q. We
denote it by M(P,Q,R).
Remark 4.4.4. This is a special case of Definition 4.3.13 which covered all Shilov
boundaries, and it specializes to Definition 2.2.14 in the Einstein universe when
n = 2.
Since Lag(R2n) is the Shilov boundary for the bounded domain realization of
the symmetric space of Sp(2n,R), it is an example of the general construction in
Section 4.3. In fact, the Maslov index we just defined agrees with the more general
version that we introduced before. Hence, the relation defined by
−−−→
PQR whenever
M(P,Q,R) = n is a partial cyclic order on Lag(R2n), enabling us to apply the
constructions and results from Section 4.2.
We also remark that the definition makes sense for any isotropic subspace Q, not
only the maximal isotropic ones.
77
Figure 4.3: Four intervals between Lagrangians in increasing order.
78
The following property of the Maslov index is well-known.
Proposition 4.4.5. The Maslov index classifies orbits of triples of pairwise trans-
verse Lagrangians, i.e. the map
(P,Q,R) 7→ M(P,Q,R)
induces a bijection from orbits of pairwise transverse Lagrangians under Sp(2n,R)
to the set {−n,−n+ 2, . . . , n}.
The Maslov index and the reflection in a pair of Lagrangians are related in the
following way:
Proposition 4.4.6.
M(P, σPQ(V ), Q) = −M(P, V,Q).
Proof.
BPQ
(
σPQ(u), σPQ(v)
)
= ω(σPQ(u), v) = −ω(u, σPQ(v)) = −BPQ(u, v).
The proposition above means that reflections reverse the partial cyclic order.
4.4.2 Fundamental domains
In the special case of Sp(2n,R), the Schottky groups we obtain admit nice fun-
damental domains for their action on RP2n−1. The domain of discontinuity which is
the orbit of this fundamental domain is in general hard to describe, but it simplifies
in some cases.
We will proceed as follows: First, we associate a “halfspace” in RP2n−1 to each
79
interval in Lag(R2n) and explain how to construct the fundamental domain. Then
we cover some preliminaries which will allow us to explicitly identify the domain
of discontinuity for generalized Schottky groups modeled on an infinite area hyper-
bolization without cusps. More specifically, we explain how to identify an interval
with the symmetric space associated with GL(n,R) and how to use a contraction
property from [Bou93] for maps sending one interval into another.
4.4.2.1 Positive halfspaces and fundamental domains
Definition 4.4.7. Let P,Q be an ordered pair of transverse Lagrangians. We define
the positive halfspace P(P,Q) as the subset
P(P,Q) := {` ∈ RP2n−1 | BPQ|`×` > 0}.
It is the set of positive lines for the form BPQ.
The positive halfspace P(P,Q) is bounded by the conic defined by BPQ = 0.
This type of bounding hypersurface was introduced by Guichard and Wienhard in
order to describe Anosov representations of closed surfaces into Sp(2n,R). They are
also the boundaries of R-tubes defined in [BP15]. A symplectic linear transforma-
tion T ∈ Sp(2n,R) acts on positive halfspaces in the following way : TP(P,Q) =
P(TP, TQ).
Proposition 4.4.8. Let P,Q be an ordered pair of Lagrangians. Then,
P(Q,P ) = P(P,Q)c = σPQ(P(P,Q))
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Proof. For the first equality,
BQP (v, w) = ω(v, σQP (w)) = ω(v,−σPQ(w)) = −BPQ(v, w).
For the second equality, notice that BPQ(σPQ(v), σPQ(w)) = −BPQ(v, w).
Proposition 4.4.9. A positive halfspace is the projectivisation of an interval, that
is,
P(P,Q) =
⋃
L∈(P,Q)
P(L)
Proof. If ` ⊂ L for some L ∈ (P,Q), then
BPQ|`×` > 0
and so ` ∈ P(P,Q).
Conversely, if ` ∈ P(P,Q), then we wish to find a Lagrangian L ⊃ ` with
M(P,L,Q) = n. Consider the subspace V = 〈`, σPQ(`)〉. The form BPQ has sig-
nature (1, 1) on that subspace, and so its orthogonal has signature (n − 1, n − 1).
Moreover, the form ω is nondegenerate on V so V ⊥ω is a symplectic subspace. Notice
that
V ⊥B = 〈`, σPQ(`)〉⊥B = `⊥B ∩ (σPQ(`))⊥B = `⊥B ∩ `⊥ω = V ⊥ω .
So we can pick a positive definite Lagrangian L′ ⊂ V ⊥, which will be orthogonal
to ` for both ω and BPQ, so L = 〈L′, `〉 is a positive definite Lagrangian containing
`.
Lemma 4.4.10. If (P,Q,R, S) is a cycle in Lag(R2n) and V ∈ Lag(R2n) such that
M(P, V,Q) = n, then M(R, V, S) = −n.
81
Proof. Using the cocycle relation,
M(V,Q,R)−M(P,Q,R) + M(P, V,R)−M(P, V,Q) = 0
so
M(V,Q,R) + M(P, V,R) = 2n
which implies that M(V,Q,R) = M(P, V,R) = n.
Similarly,
M(V,R, S)−M(P,R, S) + M(P, V, S)−M(P, V,R) = 0
so
M(V,R, S) + M(P, V, S) = 2n
which means that M(V,R, S) = n and so M(R, V, S) = −n.
Now we can prove the disjointness criterion for positive halfspaces.
Proposition 4.4.11. If (P,Q,R, S) is a cycle in Lag(R2n), then P(P,Q) is disjoint
from P(R, S).
Proof. Let ` ∈ P(P,Q). By Proposition 4.4.9, ` ⊂ L for some Lagrangian L with
M(P,L,Q) = n. By Lemma 4.4.10, M(R,L, S) = −n which means that BRS|` < 0
and so ` /∈ P(R, S).
For any generalized Schottky group, we can use this previous proposition to
construct a fundamental domain. If the defining intervals for the Schottky group
are (a±1 , b
±
1 ), . . . , (a
±
g , b
±
g ) ⊂ Lag(R2n), let
D =
g⋂
j=1
(P(a+j , b+j ) ∪ P(a−j , b−j ))C .
82
Figure 4.4: A pair of disjoint positive halfspaces in RP3
That is, D is the subset of RP2n−1 which is the complement of the positive halfspaces
defined by each interval. It is a closed subset since each positive halfspace is open.
The interiors of the translates of D are all disjoint by the two previous propositions
and the boundary components are identified pairwise, so D is a fundamental domain
for its orbit (Fig. 4.5). This orbit is in general hard to describe, but in some cases
we can identify it precisely.
In the definition of generalized Schottky subgroups, we required that the model
be a finite area hyperbolization. This is an artificial requirement which made the
analysis of maximal representations simpler. In what follows, we will assume that the
model Schottky group acting on RP1 is defined by intervals with disjoint closures, so
it corresponds to an infinite area hyperbolization. The advantage of using intervals
with disjoint closures lies in the contraction property proven in [Bou93] which we
will exploit later.
83
Figure 4.5: The first two generations of positive halfspaces for a two-generator
Schottky group in Sp(4,R).
84
4.4.2.2 Intervals as symmetric spaces
We will now describe how to identify an interval in Lag(R2n) with the sym-
metric space associated with GL(n,R), endowing any interval with a Riemannian
metric.
Let P,Q ∈ Lag(R2n) be two transverse Lagrangians. As we saw earlier in Corollary
4.3.16, all Lagrangians in the interval (P,Q) have to be transverse to Q, so they are
graphs of linear maps f : P → Q. The isotropy condition on f is given by
ω(v + f(v), v′ + f(v′)) = ω(v, f(v′)) + ω(f(v), v′) = 0 ∀v, v′ ∈ P.
Now we recall from our discussion of the Maslov index that we can associate the
bilinear form
BPQ : P ⊕Q→ R
(v, w) 7→ ω(v, σPQ(w))
to this splitting, and the index of its restriction to graph(f) is the Maslov index
M(P, graph(f), Q). We observe that this restriction is given by
BPQ(v + f(v), v′ + f(v′)) = ω(v, f(v′))− ω(f(v), v′) = 2ω(v, f(v′)),
where the last equation follows from the isotropy condition on f . This bilinear form
on graph(f) can also be seen as a symmetric bilinear form on P . Maximality of the
Maslov index then translates to this form being positive definite.
Conversely, given a symmetric bilinear form b on P , we obtain, for any v′ ∈ P , a
85
linear functional (
v 7→ 1
2
b(v, v′)
)
∈ P ∗.
Using the isomorphism
Q→ P ∗
w 7→ ω(·, w),
we see that there is a unique vector f(v′) ∈ Q such that b(v, v′) = 2ω(v, f(v′)) ∀v ∈
P . This uniquely defines a linear map f : P → Q, and
2 (ω(v, f(v′)) + ω(f(v), v′)) = b(v, v′)− b(v′, v) = 0,
so graph(f) is a Lagrangian. The Maslov index M(P, graph(f), Q) is maximal if and
only if b is positive definite. This identifies (P,Q) with the space of positive definite
symmetric bilinear forms on P , which is the symmetric space of GL(P ).
The stabilizer in Sp(2n,R) of the pair (P,Q) can be identified with GL(P ) since
any element A ∈ GL(P ) uniquely extends to a linear symplectomorphism of R2n
fixing Q: The linear forms v 7→ ω(A(v), w) on P , for w ∈ Q, give rise to a unique
automorphism A∗ : Q→ Q such that
ω(A(v), w) = ω(v,A∗(w)).
Then A⊕(A∗)−1 is the unique symplectic extension of A fixing Q; we abuse notation
slightly and denote it by A as well. It acts on graphs f : P → Q by
f 7→ AfA−1,
86
and on bilinear forms on P by
(A · b)(v, v′) = b(A−1v,A−1v′).
The identification of graphs and bilinear forms is equivariant with respect to these
actions. In particular, StabSp(2n,R)(P,Q) identifies with the isometry group of the
symmetric space (P,Q).
4.4.2.3 The Riemannian distance on intervals
Here is a simple formula for the Riemannian distance between two points in
the interval (P,Q):
Definition 4.4.12. Let f, g be linear maps from P to Q whose graphs are elements
of (P,Q). Let λ1, . . . , λn be the eigenvalues of the automorphism fg
−1. Then, define
dPQ(f, g) =
√√√√ n∑
i=1
log(λi)2.
The following useful proposition is proved in [Bou93].
Proposition 4.4.13. Let T ∈ Sp(2n,R) such that T (P,Q) ⊂ (P,Q). Then, T is a
Lipschitz contraction for the distance dPQ.
Corollary 4.4.14. Let T ∈ Sp(2n,R) such that T (P,Q) ⊂ (R, S). Then, for any
X, Y ∈ (P,Q),
dRS(TX, TY ) ≤ CdPQ(X, Y )
for some constant 0 < C < 1.
87
Let us now prove the main lemma for the description of domains of discon-
tinuity. Let ρ : Γ → Sp(2n,R) define a generalized Schottky group in Sp(2n,R).
Assume that the model Γ is defined by intervals with distinct endpoints, so that the
intervals in Lag(R2n) have disjoint closures.
Lemma 4.4.15. Let γ ∈ ρ(Γ) be a word of reduced length ` in the generators Ti
and their inverses, with first letter T and last letter S. We denote their attracting
and repelling intervals by I± and J±. Then, for any Schottky interval K 6= J− and
X, Y ∈ K,
dI+(γ(X), γ(Y )) < C
`dK(X, Y )
for some 0 < C < 1 depending only on the set of generators.
Proof. Since a generator Tk maps the interval −I−k into I+k , we can consider it as a
map from any Schottky interval L 6= I−k into I+k . All of these maps are Lipschitz
contractions by Corollary 4.4.14.
Now let C be the maximum Lipschitz constant of all such maps, for 1 ≤ k ≤ 2g.
We have
dJ+(SX, SY ) < CdK(X, Y ).
Composing contractions, we obtain
dI+(γ(X), γ(Y )) < C
`dK(X, Y ).
4.4.2.4 The domain of discontinuity
Now we analyze the orbit ρ(Γ) · D of the fundamental domain D ⊂ RP2n−1
which was defined in Section 4.4.2.1. Using Lemma 4.4.15, we first define a map
88
from the boundary of Γ, which is a Cantor set, into Lag(R2n).
Proposition 4.4.16. Let L ∈ ⋂gj=1(−I+j ∩−I−j ) where I±j are the defining intervals
of the generalized Schottky group. The evaluation map η0(γ) = ρ(γ)(L) induces an
injective map η : ∂Γ → Lag(R2n) independent of the choice of L. Moreover, η is
continuous and increasing.
Proof. Let x ∈ ∂Γ be a boundary point. Then x corresponds to a unique infinite
sequence in the generators Ti and their inverses, where this sequence is reduced in
the sense that no letter is followed by its inverse. We denote by x(k) ∈ Γ the word
consisting of the first k letters of x. Then the map η will be defined by taking the
limit
η(x) = lim
k→∞
ρ(x(k))(L).
Let us first check that this limit does in fact exist. Recall that we introduced k-th
order intervals and a bijection between words of length k and k-th order intervals
in Section 4.2. By the specific choice of L, its image ρ(x(k))(L) has to lie in the
interval Ix(k) corresponding to the word x
(k). Since the first k letters of any word
x(m), m > k agree with x(k), the intervals Ix(k) form a nested sequence. Now we
want to make use of the contraction property from the previous subsection. We first
observe that since our model uses Schottky intervals with disjoint closures, second
order intervals are relatively compact subsets of first order intervals. Let I(2) ⊂ I(1)
be such a configuration. Since the number of second order intervals is finite, there
is a uniform bound M such that
diamI(1)(I
(2)) < M,
89
where we used the metric on the symmetric space I(1). Then, denoting the first
letter of x by T , Lemma 4.4.15 tells us that
diamIT (Ixk) < MC
k−2.
This contracting sequence of nested subsets of the symmetric space IT thus has a
unique limit, and η is well-defined. By the same argument, we see that this limit
does not depend on the choice of L.
We now show continuity of η. Let yn → x be a sequence in ∂Γ converging to
x. This implies that for any N ∈ N, we can find n0 such that for all n ≥ n0, the
first N letters of yn and x agree. In this situation, η(yn) and η(x) lie in the same
interval Ix(N) and so we conclude, if the first letter of x is T , that
dIT (η(x), η(yn)) < MC
N−2.
Finally, we prove positivity in a similar way to Theorem 4.2.7. For any x, y, z ∈
∂Γ such that −−→xyz (where we use the natural embedding of ∂Γ in S1 to get the
cyclic order) we can find a large enough K so that IxK ,IyK and IzK have disjoint
closures. But since the cyclic relations on k-th order intervals are the same in S1
as in Lag(R2n), for any P ∈ IxK , Q ∈ IyK , R ∈ IzK we have
−−−→
PQR. In particular,
−−−−−−−−→
η(x)η(y)η(z).
Remark 4.4.17. The map η that we define is related to the map ξ of Theorem
4.2.7. In this case, the endpoints of k-th order intervals are not dense in S1, so we
cannot get a map on the whole circle. However, because the intervals have disjoint
closures, we get continuity on both sides rather than just left-continuity.
90
The next lemma relates the construction of the limit map η with the positive
halfspaces that intervals define in RP2n−1.
Lemma 4.4.18. Let Lk1, L
k
2 be sequences of Lagrangians such that L
k
1 → L and
Lk2 → L with
−−−−→
Lk1LL
k
2 for all k. Then,
∞⋂
k=1
P(Lk1, Lk2) =
∞⋂
k=1
P(Lk1, Lk2) = P(L).
Proof. Assume BLk1Lk2 (v, v) ≥ 0 for all k. Then we can find vk
k→∞−−−→ v such that
BLk1Lk2 (vk, vk) > 0 for all k. Now, by Proposition 4.4.9, vk can be completed to a
Lagrangian Lk with M(Lk1, L
k, Lk2) = n, so L
k ⊂ (Lk1, Lk2) for all k, which implies
Lk → L, and so v ∈ L.
Now we are ready to describe the orbit ρ(Γ)D.
The union of D with the positive halfspaces defining the Schottky group is all
of RP2n−1, by definition of D. Denote by Γ` the set of words in Γ of length up to `.
Then, the union of ρ(Γ`)D with the projectivizations (positive halfspaces) of all `-th
order intervals again covers all of RP2n−1. Thus, when taking words of arbitrary
length in Γ, these two pieces become respectively the full orbit ρ(Γ)D and limits of
nested positive halfspaces, which by Lemma 4.4.18 collapse to the projectivization
of a single Lagrangian. We conclude:
Theorem 4.4.19. The orbit ρ(Γ)D is the complement of a Cantor set of projec-
tivized Lagrangian n-planes in RP2n−1. This Cantor set is exactly the projectivization
of the increasing set of Lagrangians defined by the boundary map η.
Remark 4.4.20. The symplectic structure on R2n induces a contact structure on
91
RP2n−1 preserved by the symplectic group. The projectivizations of Lagrangian sub-
spaces correspond to Legendrian (n− 1)-dimensional planes in RP2n−1.
4.5 Oriented flags in three dimensions
The theory of positive configurations of flags of Fock and Goncharov [FG06]
hints at the existence of a partial cyclic order on the space of flags in Rn.
Since positivity of triples of flags is preserved under all permutations, we have
to look at oriented flags. The space of oriented flags in Rn admits a partial cyclic
order, with some care needed when n is even. For ease of exposition, we describe
this ordering for n = 3 and the resulting Schottky groups. We treat the general case
in the work in progress [BT17].
4.5.1 Hyperconvex configurations
Definition 4.5.1. An oriented flag in R3 is a sequence of subspaces ` ⊂ P ⊂ R3
together with a choice of orientation on each subspace. We will sometimes denote
and oriented flag by the pair (`, P ).
Definition 4.5.2. A basis for an oriented flag (`, P ) is an oriented basis e1, e2, e3 of
R3 such that e1 is an oriented basis for ` and e1, e2 is an oriented basis for P .
If we fix an oriented basis B of R3, we can also denote an oriented flag F by a
3× 3 matrix whose columns are the coordinates in the basis B for a flag basis of F .
We denote by F the space of oriented flags.
Fix a choice of orientation on R3. Throughout this section, we will be consid-
92
ering oriented vector spaces. The direct sum operation can be applied to oriented
vector spaces, with the orientation on the sum being given by the concatenation of
two oriented bases.
Definition 4.5.3. A pair of oriented flags (`, P ), (`′, P ′) is called transverse if the
following condition holds :
`⊕ P ′ = P ⊕ `′ = R3,
where equality is understood in terms of oriented vector spaces.
Proposition 4.5.4. Transversality of oriented flags is symmetric. That is, if F1, F2
are transverse, then F2, F1 are transverse.
Proof. Let e1, e2, e3 and e
′
1, e
′
2, e
′
3 be oriented bases for F1, F2, respectively. Assume
F1, F2 are transverse. Then, F2, F1 are transverse if the following two bases are
oriented :
e′1, e2, e3,
e′2, e
′
3, e1.
But these bases are cyclic permutations of e2, e3, e
′
1 and e1, e
′
2, e
′
3 which are oriented
bases by transversality of F1, F2.
Example 4.5.5. Let e1, e2, e3 be the canonical oriented basis of R3. The oriented
flags ([e1], [e1]⊕ [e2]) and ([e3], [e2]⊕ [e3]) are transverse. If we switch the orientation
on the 2-plane of the second flag, they are not considered transverse anymore since
the orientation given by (e1, e3, e2) does not coincide with the fixed orientation.
93
Figure 4.6: A hyperconvex configuration of three oriented flags, projected to the
2-sphere of directions.
Definition 4.5.6. A triple of oriented flags (F1, F2, F3) is called hyperconvex if
F1, F2, F3 are pairwise transverse and the following equality of oriented vector spaces
holds :
`1 ⊕ `2 ⊕ `3 = R3.
The group SL(3,R) acts on F preserving transversality and hyperconvexity of
triples, since it consists of orientation-preserving linear transformations.
Proposition 4.5.7. SL(3,R) acts transitively on pairs of transverse flags in F .
Proof. Let F1, F2 be a pair of transverse flags. Let A1, A2 be 3 × 3 matrices rep-
resenting F1, F2 in the standard basis. Since det(A1), det(A2) > 0 we can multiply
by a positive scalar and assume A1, A2 ∈ SL(3,R). Multiply the pair by A−11 to get
I, A−11 A2.
94
Denote B = A−11 A2. The stabilizer of the flag I is the set of upper triangular
matrices with positive entries on the diagonal. Transversality implies that there is
a choice of basis for the flag F2 which makes B a matrix of the form
a b 1
c −1 0
1 0 0

which we can send using a unipotent upper triangular matrix to the standard
0 0 1
0 −1 0
1 0 0
 .
We will denote this standard matrix by I¯.
We now show that hyperconvexity gives a partial cyclic order on the set of
oriented flags in R3. Cyclicity and asymmetry follow from the fact that permuting
a basis changes its orientation according to the sign of the permutation. It remains
to show that this order is transitive.
Proposition 4.5.8. Let
A =

1 a b
0 1 c
0 0 1
 .
The triple of flags given by matrices I, AI¯, I¯ is hyperconvex if and only if a, b, c > 0
and ac− b > 0. Such a matrix A is called totally positive.
95
Proof. The transversality of the second and third flags is equivalent to the positivity
of the determinants
det

0 b −a
0 c −1
1 1 0
 = ac− b
det

0 0 b
0 −1 c
1 0 1
 = b.
The hyperconvexity of the triple is then equivalent to the positivity of the determi-
nant
det

1 b 0
0 c 0
0 1 1
 = c.
The condition a > 0 is redundant and included for symmetry.
A simple calculation shows the following proposition.
Proposition 4.5.9. The set of totally positive matrices forms a subsemigroup of
the group of unipotent upper triangular matrices.
Proposition 4.5.10 (Transitivity). If (F1, F2, F3) is hyperconvex and (F1, F3, F4)
is hyperconvex, then (F1, F2, F4) is hyperconvex.
Proof. Without loss of generality, F1 is given by the identity matrix and F3 is given
by I¯. By transversality of F1, F2 and F1, F4 we can write F2 = AI¯ and F4 = BI¯
where A,B are unipotent upper triangular matrices.
96
Since (F1, F2, F3) is hyperconvex, Proposition 4.5.8 shows that A is totally pos-
itive. Similarly, since (F1, F3, F4) is hyperconvex, we know that (I, B
−1I¯ , I¯) is hy-
perconvex and so B−1 is totally positive. Since totally positive matrices form a semi-
group, B−1A is also totally positive, which means that (I, B−1AI¯, I¯) is hyperconvex.
Left multiplying by B throughout, we conclude that (I, AI¯, BI¯) = (F1, F2, F4) is hy-
perconvex.
As so we have proven
Theorem 4.5.11. The hyperconvexity relation on triples of oriented flags is a partial
cyclic order. We will denote it by
−−−−→
F1F2F3 as in the previous sections.
We proved in Proposition 4.5.8 that the interval between two oriented flags
(F1, F2) can be identified with the open set of totally positive, unipotent, upper
triangular 3 × 3 matrices. These sets are homeomorphic to balls in the space of
oriented flags.
By similar arguments to those of Theorem 4.3.19, we obtain the following:
Theorem 4.5.12. The PCO on the space of oriented flags F is increasing-complete,
full and proper (it satisfies the hypotheses of Theorem 4.2.7).
97
Bibliography
[BBH+16] G. Ben Simon, M. Burger, T. Hartnick, A. Iozzi and A. Wienhard,
On order preserving representations, ArXiv e-prints (January 2016),
1601.02232.
[BCD+08] T. Barbot, V. Charette, T. Drumm, W. M. Goldman and K. Melnick,
A primer on the (2 + 1) Einstein universe, in Recent developments in
pseudo-Riemannian geometry, ESI Lect. Math. Phys., pages 179–229,
Eur. Math. Soc., Zu¨rich, 2008.
[BCDG14] J.-P. Burelle, V. Charette, T. Drumm and W. Goldman, Crooked half-
spaces, Enseign. Math. 60(1), 43–78 (2014).
[BCFG17] J.-P. Burelle, V. Charette, D. Francoeur and W. Goldman, Einstein
tori and crooked surfaces, ArXiv e-prints (February 2017), 1702.08414.
[BIW10] M. Burger, A. Iozzi and A. Wienhard, Surface group representations
with maximal Toledo invariant, Ann. of Math. (2) 172(1), 517–566
(2010).
[Bou93] P. Bougerol, Kalman filtering with random coefficients and contrac-
tions, SIAM J. Control Optim. 31(4), 942–959 (1993).
[BP15] M. Burger and M. B. Pozzetti, Maximal representations, non
Archimedean Siegel spaces, and buildings, ArXiv e-prints (September
2015), 1509.01184.
[BT16] J.-P. Burelle and N. Treib, Schottky groups and maximal representa-
tions, ArXiv e-prints (September 2016), 1609.04560.
[BT17] J.-P. Burelle and N. Treib, Examples of generalized Schottky groups
and connected components, In Preparation (2017).
98
[CDGM03] V. Charette, T. Drumm, W. Goldman and M. Morrill, Complete flat
affine and Lorentzian manifolds, Geom. Dedicata 97, 187–198 (2003),
Special volume dedicated to the memory of Hanna Miriam Sandler
(1960–1999).
[Cec08] T. E. Cecil, Lie sphere geometry, Universitext, Springer, New York,
second edition, 2008, With applications to submanifolds.
[CFLD14] V. Charette, D. Francoeur and R. Lareau-Dussault, Fundamental do-
mains in the Einstein universe, Topology Appl. 174, 62–80 (2014).
[Cle04] J.-L. Clerc, L’indice de Maslov ge´ne´ralise´, J. Math. Pures Appl. (9)
83(1), 99–114 (2004).
[DGK14] J. Danciger, F. Gue´ritaud and F. Kassel, Fundamental domains for free
groups acting on anti-de Sitter 3-space, ArXiv e-prints (October 2014),
1410.5804.
[Dru92] T. A. Drumm, Fundamental polyhedra for Margulis space–times,
Topology 31(4), 677–683 (1992).
[DT87] A. Domic and D. Toledo, The Gromov norm of the Kaehler class of
symmetric domains, Math. Ann. 276(3), 425–432 (1987).
[FG06] V. Fock and A. Goncharov, Moduli spaces of local systems and higher
Teichmu¨ller theory, Publ. Math. Inst. Hautes E´tudes Sci. (103), 1–211
(2006).
[FK94] J. Faraut and A. Kora´nyi, Analysis on symmetric cones, Oxford
Mathematical Monographs, The Clarendon Press, Oxford University
Press, New York, 1994, Oxford Science Publications.
[Fra03] C. Frances, The conformal boundary of Margulis space-times, C. R.
Math. Acad. Sci. Paris 336(9), 751–756 (2003).
[Fra05] C. Frances, Lorentzian Kleinian groups, Comment. Math. Helv. 80(4),
883–910 (2005).
[Gol80] W. M. Goldman, Discontinuous Groups and the Euler Class, ProQuest
LLC, Ann Arbor, MI, 1980, Thesis (Ph.D.)–University of California,
Berkeley.
[Gol15] W. M. Goldman, Crooked surfaces and anti-de Sitter geometry, Geom.
Dedicata 175, 159–187 (2015).
[GW12] O. Guichard and A. Wienhard, Anosov representations: domains of
discontinuity and applications, Invent. Math. 190(2), 357–438 (2012).
99
[Kan91] S. Kaneyuki, On the causal structures of the Silov boundaries of sym-
metric bounded domains, in Prospects in complex geometry (Katata
and Kyoto, 1989), volume 1468 of Lecture Notes in Math., pages 127–
159, Springer, Berlin, 1991.
[Nor86] M. V. Nori, The Schottky groups in higher dimensions, in The Lef-
schetz centennial conference, Part I (Mexico City, 1984), volume 58 of
Contemp. Math., pages 195–197, Amer. Math. Soc., Providence, RI,
1986.
[Nov82] V. Nova´k, Cyclically ordered sets, Czechoslovak Math. J. 32(107)(3),
460–473 (1982).
[SV03] J. Seade and A. Verjovsky, Complex Schottky groups, Aste´risque (287),
xx, 251–272 (2003), Geometric methods in dynamics. II.
[Tit72] J. Tits, Free subgroups in linear groups, J. Algebra 20, 250–270 (1972).
[Tol79] D. Toledo, Harmonic maps from surfaces to certain Kaehler manifolds,
Math. Scand. 45(1), 13–26 (1979).
[Tol89] D. Toledo, Representations of surface groups in complex hyperbolic
space, J. Differential Geom. 29(1), 125–133 (1989).
[Wie04] A. K. Wienhard, Bounded cohomology and geometry, Bonner Mathe-
matische Schriften [Bonn Mathematical Publications], 368, Universita¨t
Bonn, Mathematisches Institut, Bonn, 2004, Dissertation, Rheinische
Friedrich-Wilhelms-Universita¨t Bonn, Bonn, 2004.
100