Workshop Boundaries of groups and representations, Vienna, february 22-24, 2016
1. Alexander Lubotzky: Arithmetic quotients of the mapping class group
Joint work with Fritz Grunewald (GAFA 2008, automorphism group of free groups), Michael Larsen, Justin Malestein (GAFA 2015).
Let , mapping class group, means either one of them. maps onto and maps to , these are arithmetic quotients, but there are many more.
Theorem 1 For every pair where is a finite group with generators and an irreducible -representation of , there exists an explicitely given arithmetic group and a virtual epimorphism , where virtual means that is defined on a finite index subgroup of and is onto a finite index subgroup of .
The classical case correspond to .
If is a finite group and is an epimorphism, , if is viewed as an -module, then the -module is isomorphic to (Gashitz). A similar result holds for surface groups : is isomorphic to (Chevalley-Weil).
Write as a sum of simple -algebras
with , where is a finite extension of and a central division algebra.
In both cases, define
This is the set of automorphisms preserving (viewed as a finite index subgroup in rather than in for the moment). Then preserves and as an -module. Since has a (resp. ) summand, we get a morphism (resp. ). I am interested in the other summands (resp. ).
1.2. Main technical theorem
The image of in preserves a lattice, since we tensored a -module into a -module. Hence it is contained in the points of a -algebraic group . In the mapping class group case, the modules inherit a bilinear form arising from cup-products in .
If or is degenerate, meaning, in the free group case, mapping one of the basis to the identity
Let us start with the simplest example . Then has index 2 in , has rank , whence 2 representations, one, the classical one, to , and a new one, to . For a while, I thought the image was nilpotent, this would have solved the irritating question wether has property (T). But I was wrong, we get a virtual epimorphism of to .
For , we get a virtual epimorphism to . I think this should not indicate that does not have property (T) for higher .
Let . Then where is the cyclotomic field. This yields an epimorphism of to .
There is some theory about which division algebras arise as summands in group algebras (the Schur subgroup of the Brauer group). This helps. Here, we get a virtual epimorphism of to for each using .
The mapping class group case requires dealing with sesquilinear forms over noncommutative rings. The involution arises from on . Over the reals, there are only three central divison algebras, , and , the arithmetic groups that we encounter are types , , the ring of integers of , types where , and types … but we found no type .
The free group case could be handled more or less by hand, using Fox calculus.
The mapping class group case is harder. The handlebody subgroup (Hamenstädt) of the mapping class group provides an action of in the mapping class group picture, and this helped a lot.
What are the kernels of these epimorphisms ? What is their intersection ?
2. Ludovic Marquis: projectivization of some Dehn fillings on hyperbolic 4-orbifolds
Joint work with Suhyoung Choi and Gye-Seon Lee.
2.1. Coxeter groups
We deal with Coxeter groups. Given a graph with labels , consider the presentation , if no edge joining to , otherwise. This produces a group .
Coxeter, Vinberg: if is irreducible (connected graph), then is either finite, either virtually (both these cases are classified) or pretty large.
2.2. Andreev’s theorem
Let be 3-dimensional polyhedron. Label each edge with an angle . Andreev’s theorem realizes as a convex hyperbolic polyhedron with dihedral angles , in a unique manner (up to hyperbolic motions). If are divisors of , reflections in the faces of generate a Coxeter group. Faces of correspond to vertices of the Coxeter graph, certain edges of to edges of the graph.
2.3. Dehn filling
What if one deforms by collapsing an edge ? The Coxeter graph persists. Let label the vanishing edge with angle . All Coxeter groups are quotients of , hence representations . As tends to infinity, converge to .
This behaviour never arises in higher dimensions.
Theorem 2 (Garland-Raghunathan) For all , for all lattices , local rigidity holds: every representation of in a neighborhood of the obvious one is conjugate to it.
Nevertheless, we shall perform Dehn filling in projective geometry.
2.4. Reflection groups in projective geometry
Here is a projective variant of Poincaré’s theorem on hyperbolic reflection groups.
Theorem 3 (Tits-Vinberg) Let be a polytope of . For each facet of , pick a reflection fixing pointwise, in such a way that each time faces and intersect, is the direct sum of identity on and a 2-dimensional rotation in a complementary plane. Then
- generate a Coxeter group with a discrete and faithful action on projective space.
- is a convex subset of projective space.
- The interior of is tiled by the .
- Theaction of is proper with as a fundamental domain.
- Assume that is irreducible and large, and that the action of on is irreducible. Then is properly convex.
2.5. Projective Dehn filling
We study an example of a graph , an edge of which is labelled . describes a polyhedron which is the product of two triangles.
Theorem 4 For finite, there are two polytopes realizing . For , there is only one, congruent to a hyperbolic polytope.
It is standard that, for such examples,
- For finite , acts cocompactly on . is not tructly convex, its boundary is not .
- For , is an ellipsoid and has finite volume.
In our examples, contains properly embedded triangles (i.e. ) but no higher dimensional tetrahedra (unlike previously known similar examples by Benoist).
Extreme points in the boundary ?
What about attaching handles in other intermediate dimensions ?
Bounds on dimensions of projective reflection groups ? In principle, such a bound should exist.
3. Gye-Seon Lee: Collar Lemma for Hitchin representations
The Collar Lemma for hyperbolic syrfaces (Linda Keen) implies that if two curves intersect, there is a lower bound for the length of one in terms of the length of the other, independently of the hyperbolic structure on the surface. We generalize this to other families of representations.
3.1. Hitchin representations
Let be the space of conjugacy classes of representations which are discrete, faithful and cocompact on a convex open subset of projective plane. I.e. the space of convex real projective structures on .
Let be the Hitchin component. Then Teichmüller space and .
Labourie : Hitchin representations are discrete and faithful, and every element is mapped to a diagonalizable matrix whose eigenvalues have distinct absolute values. We denote by . Then equals hyperbolic length of closed geodesics if and the Hilbert length of closed geodesics if .
Linda Keen’s result states that a short geodesic has a wide tubular neighborhood which is a topological annulus. It implies that if and have nonzero intersection number , then
Theorem 5 (Zhang-Lee) Let belong to the Hitchin component. Same statement holds, where is replaced with .
This does not hold for all Anosov representations.
Example 1 Quasi-Fuchsian representations in form a product of 2 copies of Teichmüller space. By Epstein-Marsden-Markovic, is bounded above by twice the min of the lengths of coordinates, which contradicts a Collar Lemma.
Linda Keen’s result is sharp (consider punctured tori). Recently, Nicolas Tholozan obtained a sharp generalization for . Our result is not sharp.
Corollary 6 For a Hitchin representation, there are at most primitive closed curves with length .
If , this is nice projective geometry. Ratios of lengths are interpreted as cross ratios of 4-tuples of points or lines. The key step is to control the order of points (axes of group elements and their images) along the ideal boundary.
In general, we use Labourie’s realization of the ideal boundary of as a completely positive curve in the flag manifold. Notions like being on the positive side of a flag, or cross-ratios geenralize to the flag manifold.
Tholozan proves that every convex real projective structure is dominated by a Fuchsian one, in the sense that the length function is uniformly smaller. We do not know wether a sharp bound can hold for higher .
4. Joan Porti: Geometry and dynamics of Anosov representations I
Joint work with Misha Kapovich and Bernhard Leeb.
4.1. Anosov representations
The notion appears for surface groups and , in his study of Hitchin’s component (2006).
The general definition, for hyperbolic groups and semi-simple Lie groups, is due to Olivier Guichard and Anna Wienhard (2012).
Definition 7 Let be a symmetric space of noncompact type, a Borel subgroup, . Let be a hyperbolic group, let . A boundary embedding is an injective continuous equivariant map . It is antipodal (resp. generic) if pairs of distinct points are mapped to antipodal pairs (resp. pairs in general positions) in .
Note that existence of a boundary embedding iplies that is discrete with finite kernel.
Definition 8 Say is Anosov if
- it admits a boundary embedding ,
- let be a geodesic ray in emanating from the identity element. Then there are uniform constants and such that
We can prove that uniformity of constants is not crucial here.
A symmetric space has a chamber valued distance, inspired by the theory of Coxeter groups. Indeed, is a Weyl chamber, i.e. a convex polygonal sector in .
Say a discrete subgroup of is regular if the distance of the chamber point to the boundary of the chamber tends to infinity as tends to infinity in . I.e. long segments joining orbit points tend to regular geodesics.
Say a discrete subgroup of is regular if the distance to the boundary is bounded below by an affine function of . This allows to define a visual limit set in the visual boundary which is in the regular part (the union of interiors of chambers in the Tits building structure).
4.3. Coarse geometry
Say a finitiely generated subgroup of is undistorted if orbit map is a quasi-isometric embedding.
Labourie and Guichard-Wienhard showed that Anosov representations are uniformly regular and undistorted. We prove the converse.
Theorem 9 hyperbolic, semi-simple. The following are equivalent:
- is (non-uniformly) Anosov.
- is uniformly regular and undistorted (URU).
- is Morse.
Morse means that that long geodesic segments of are mapped into uniform neighborhoods of diamonds. A diamond is a segment in the sense of the chamber valued distance: the intersection of two Weyl chambers in the same maximal flat, pointing in opposite directions.
Morse property can be verified locally : on large enough balls.
Morse implies Anosov. A sequence of diamonds staying a bounded distance away from each other converge to a Weyl chamber.
Morse implies URU. Every moves an apex along a maximal flat a definite amount, this implies that quasi-geodesics are mapped to quasi-geodesics.
Anosov implies Morse.
URU implies Morse. Use contraction properties of the projection to a maximal flat: uniformly regular quasi-geodesics are uniformly close to diamonds.
4.5. An application: construction of Schottky groups
Let and be axial geodesics, with regular axes and pair-wise generic end-points. Then, for large enough , , and generate a free, URU subgroup. This is a substitute for ping-pong, which does not seem to work easily in higher rank.
Benoist uses a ping-pong argument somewhere. So do Breuillard and Gelander.
5. Bernhard Leeb: Geometry and dynamics of Anosov representations II
Joint work with Misha Kapovich.
I explain how the previous coarse considerations lead to study Finsler metrics on symmetric spaces, and compactifications that have good dynamical properties.
5.1. Finsler metrics
Pick a regular vector in the model Weyl chamber. This defines a linear function on the Weyl chamber, whence a Weyl group-invariant norm on the model maximal flat, whence a -invariant metric on the symmetric space. It turns out that, given points and , the diamond between them coincides with the union of all Finsler geodesics joining them.
A proper geodesic metric space embeds into continuous functions mod additive constants , via . The closure of the image is compact. is called the horoboundary of .
If tends to infinity, to what does mod constant converge to ? If, seen from , stays inside the Weyl chamber, the limit is a Busemann function which is linear, associated to , in Weyl chambers. If approaches a singular ray, the limit is the max of two Busemann functions. In other words, compared to the visual (Tits) boundary,
- interiors of visual chambers are collapsed to points of the horoboundary.
- vertices of visual chambers are blown up into cells.
5.3. Properties of the horoboundary
Theorem 10 The -equivariant compactification
- is independent of the chosen Finsler metric,
- has finitely many -orbits corresponding to conjugacy classes of parabolic subgroups of , or equivalently, to simplices of the Tits boundary.
- has the structure of a manifold with corners,
- is homeomorphic to a ball,
- coincides with the maximal Satake compactification.
The largest (open) stratum is , corresponds to the empty simplex. The smallest (closed) stratum is , corresponding to maximal Weyl chambers. For intermediate simplices , consider the space of strong asymptotic classes of Weyl sectors asymptotic to .
Theorem 11 Let be a discrete subgroup of .
- There exist natural saturated domains such that acts properly discontinuously. This provides us with a bordification of as an orbifold with corners.
- Anosov is cocompact on .
The converse holds.
Theorem 12 Let be a discrete subgroup of . Assume that is unformly regular and acts cocompactly on , then is Anosov.
is obtained by removing a thickening of the limit set .
5.5. An application to convergence actions
Let be a hyperbolic group, with a convergence action on a compact metrizable space . Assume that the action on the limit set is conjugate to the action of on its ideal boundary. Assume that the complement is path connected. Then is cocompact on .
This partially solves a question of Haissinsky.
6. Pierre-Emmanuel Caprace: Linear representations of lattices in Euclidean buildings
Joint work with Uri Bader and Jean Lecureux.
Keep in mind the following 3 pictures – a Euclidean plane tiled with equilateral triangles, – the incidence graph of the smallest projective plane, , a bi-partite graph with 14 vertices. – a bi-colored tree.
6.1. Buildings and lattices
Here is a characterization of -buildings.
Theorem 13 (Charney-Lytchak) An -building is a simply connected 2-dimensional simplicial complex whose links are incidence graphs of projective planes.
Today, I will call -lattice a pair where is a locally finite -building and a discrete cocompact group of automorphisms of .
Examples. the Bruhat-Tits building of where is a division algebra over a local field with finite residue field, a cocompact arithmetic lattice (I call these the classical -lattices).
Example 2 is a cocompact lattice in .
There are non-classical -buildings, I call them romantic.
Start with a finite 2-complex satisfying the link condition. Then is an -lattice.
This can be implemented by a computer search. Non-trivial issue: determine wether is classical or romantic.
Example 3 is a romantic -lattice.
It was discovered by Ronan and Tits in 1984, the presentation is due to Essert in 2011.
An enumeration of -lattices with only one vertex was performed by Cartwright-Mantero-Steger-Zappa in 1994. In 1996, Barre found a new example. Essert continued Nicolas Radu recently beat the record, with an example where links are non-Desarguesian. Its thickness (degree of the link) is 10, which is the largest among known romantic examples. It is believed that romantic
6.3. Neo-classical examples ?
Let be the Bruhat-Tits building of . A theorem of Tits guarantees that
A lattice is called Galois if it has infinite image in . Such lattices can exist only if .
For instance, if , , known as the Nottingham group, is a huge pro-finite group (it contains copies of all pro- groups).
Question. Does there exist a neo-classical lattice, i.e. a pair of a Bruhat-Tits building and a Galois lattice.
All -lattices share the following properties:
- is , is finitely presented.
- has property (T) (Pansu-Zuk 1996).
- is just-infinite (Shalom-Steger 2006).
6.5. A characterization of classical buildings
Theorem 14 Given an -lattice, the following are equivalent:
- has a linear representation with infinite image in , any field.
- is the Bruhat-Tits building of over , is arithmetic, therefore virtually contained in .
Therefore romantic or Galois lattices are not linear.
6.6. Reduction step
One can replace with where isa simple algebraic group over a local field and is Zariski-dense. This uses Tits’ trick and property (T).
6.7. Transcendental step
Here, we use Bader-Furman’s Gate theory.
Let be a countable group, a standard Borel space with a measure , quasi-preserved by . Gate theory associates to a linear reprentation a continuous representation for any polish group with an action on commuting with . is a sub-quotient of .
Note that need not be contained in . I explain what is in our setting. It is the projectivity group of the visual boundary.
6.8. Hilbert’s axiomatics for geometry
According to Hilbert 1897, a projective plane is a pair where (the set f lines) is a collection of subsets of the set of points, satisfying 3 axioms,
- 2 distinct points lie on a unique line.
- 2 distinct lines intersect in a unique point.
- there exists 4 points, no 3 of which are collinear.
The classical, Desarguesian, examples are , a field or a skew field.
Hilbert discovered an infinite non-Desarguesian plane. A few years later, finite examples were found.
A projectivity is a permutation of a line produced by an alternating chain of points and lines.
Definition 15 Let be a projective plane. We fix a line and denote by its group of projectivities.
Fact. is 3-transitive.
For instance, if , then . For finite non-Desarguesian planes, is the full symmetric group.
6.9. on the boundary
In the same way as links in projective planes are lines, a building has a visual boundary which is a spherical building (appartments are 6-cycles). The idea of a projectivity generalizes. A closed chain of opposite vertices at infinity produces a permutation of the set of (singular) geodesics converging to the starting vertex. This set is a lower rank building, the group of permutations obtained is denoted by .
For classical buildings, is the Levi part of a parabolic subgroup of .
For -buildings, a set of parallel singular geodesics is a regular tree , . We let and act on the space of geodesics, i.e. . Let be the closure of some orbit of . This turns out to admit an invariant measure, which is ergodic under .
7. Damian Osajda: Gromov boundaries with the combinatorial Loewner property
Joint work with Antoine Clais.
7.1. Loewner property
Analysis on ideal boundaries of hyperbolic groups is powerful when ideal boundary has Loewner property.
Loewner property requires a -Ahlfors regular metric which has plenty of rectifiable curves. Plenty means that condensers ( and are disjoint continua) have finite and positive capacity, and that this capacity is of the order of
Loewner property turns out to be quasi-Möbius invariant only a posteriori.
Only a small list of examples have this property. Bonk and Kleiner introduced a variant which is genuinely quasi-Möbius invariant, and coined it the combinatorial Loewner property (CLP).
7.2. Combinatorial Loewner property
A -approximation of a compact metric space is a covering by open sets which contain disjoint balls of radius and are contained in the -larger concentric balls.
Given a -approximation and positive function on , a -length is defined for curves by summing values of on pieces which intersect the curve. A -mass is defined by summing over all pieces of the covering.
Given a family of curves, minimizing -mass over functions which give -length to each curve of yields the -modulus .
Definition 16 (Bonk-Kleiner) Say a compact metric space has the combinatorial Loewner property if for all pairs of disjoint continua and , the family of curves joining to satisfies
Kleiner conjectures that if the ideal boundary of a hyperbolic group has the CLP, then its quasi-Möbius gauge contains a Loewner metric.
Examples of groups whose boundaries have CLP are still rare. I describe a new one. It is a right-angled Coxeter group.
The datum is finite graph, encoding pairs of involutions which are required to commute. The corresponding Coxeter group is hyperbolic iff the graph contains no cycles of length .
Theorem 17 Consider the equilateral triangulation of the 2-torus whose combinatorial systole is , . Take its 1-skeleton. The corresponding Coxeter group has CLP.
From Antoine Clais’ work, it follows that is a lattice of automorphisms of the associated 3-dimensional hyperbolic building.
The ideal boundary is homeomorphic to the Pontrijagin surface. Start with a 2-sphere, triangulate it, perform a connected sum with a 2-torus inside each face. triangulate again the obtained surface, perform connected sums, and so on, infinitely many times. It is a theorem of Jakobsche that the resulting topological space does not depend on choices. Fisher used it to describe ideal boundaries of cubical complexes
Indeed, the -sphere in the building is obtained from the -sphere by connect-summing 2-tori, one each time a vertex is passed.
The graph admits involutions which stabilize 2 cycles, one being pointwise fixed, the other not. We use pairs of such involutions, in order that fixed point sets disconnect the torus. This plays a role in the proof of CLP. Indeed, on , one needs to produce large families of curves following a given curve.
8. Anatoly Vershik: The absolute boundary of random walks on graphs and groups
On the classical subject of random walks on trees, I have a seemingly new result. Previously, I will explain a new conception of boundary, inspired by old papers by Dynkin. He used the words exit boundary and entrance boundary, which is a but improper, since there is no difference between exit and entrance. I prefer the word absolute.
8.1. Co-transition probabilities
Let be an oriented graph. For each vertex , a probability distribution on outgoing edges is given. This equipment suffices to define the random walk.
One can also be given co-transition probabilities, i.e. conditional probabilities on incoming edges. The random walk is a measure on the space of paths, which detrmines co-transition probabilities, but the converse is not true.
Problem. Find all probability measures on the space of paths compatible with a given set of co-transition probabilities.
Definition 18 The absolute of is the list of all ergodic measures on compatible with .
There is a symmetric (in fact identical) problem of determining all ergodic measures on compatible with a given set of transition probabilities.
Example 4 Usual random walk on .
All paths through a point have the same co-transition probabilities (a power of ). So we must determine probability measures on giving uniform conditional measures (“central measure”). In other terms, a measure invariant under permutations of vertices. Only Bernoulli measures satisfy this. So the absolute is a interval .
Our problem is expressible into operator algebra terms: characters of limits of finite…
Example 5 Discrete abelian groups.
I do not yet know the answer, but I am close to the solution.
8.2. Connection with Martin boundary
Absolute does not change if vertices are removed and their edges a replaced with direction connections. This operation changes Martin boundary.
Let denote the -simplex. The map . The absolute is the set of extremal points of the inverse limit, which, according to Choquet, is again a simplex. It is contained in Martin boundary.
8.3. Random walks on trees
Consider simple uniform random walk on regular tree of degree . The boundary is the set of ends.
Define the dynamic graph that records all paths (at level , all vertices that can be reached after steps from the initial vertex).
We know that with probability 1, the trajectory reaches an end. This gives the ergodic decomposition of our set of trajectories. Equivalently, bounded harmonic functions correspond to functions on the space of ends. This is called Poisson boundary.
The conditional Markov process for which almost all trajectories reach a given point has the same co-transition probabilities. Does this provide all Markov processes in the absolute ?
Let , . At each vertex, there is an edge pointing towards . Put probability on this edge, and uniform probabilities on the other. This defines transition probabilities, hence a Markov process, compatible with given (uniform) co-transition probabilities. It is ergodic. can be interpreted as a drift, with speed . Only if does the walk converge to . For , the speed equals to zero, i.e. distance to origin increases at sublinear speed (in fact, ).
In the 1970’s, Stanislav Molchanov solved my problem in a different language, in terms of minimal non-negative eigenfunctions of the Laplacian. If is an eigenvalue, the corresponding value of is
A phase transition occurs at . If , there is no ergodicity, trajectories are free to converge to any endpoint.
I begin having results for abelian groups, and some hope for nilpotent group. The picture is always the same: absolute fibres over Poisson boundary, with interval fibres.
9. Fanny Kassel: Proper affine actions for right-angled Coxeter groups
Joint work with Jeff Danciger and Francois Gueritaud.
There are still a number of open questions about affine manifolds. Auslander’s conjecture asserts that compact affine manifolds have solvable fundamental groups (partial answers: Fried-Goldman in dim 3, Abert-Goldman-Margulis in dim 4, under extra assumptions). At some point, it was not even clear wether compactness was necessary, until Margulis found free counterexamples in dimension 3.
Theorem 20 Every right-angled Coxeter group (RACG) with generators admits a proper action on .
Many classes of groups embed in right-angled Coxeter groups: right-angled Artin groups (Davis-Januskiewicz), virtually special groups (Haglund-Wise) as well, this includes all Coxeter groups, surface groups, fundamental groups of hyperbolic 3-manifolds.
9.1. General setting
Let be a discret group. Let be a Lie group, with the action of . The infinitesimal version of this action is an affine action of on the Lie algebra . This suggests starting from pairs of representations of , and move to pairs of a representation and a Lie algebra cocycle, since cocycles correspond to the tangent space of .
9.2. General principle: uniform contraction implies properness
2 years ago, we made the following observation. Start with a proper isometric action of on hyperbolic space . Let be another action which is uniformly contracting with respect to . Then the pair is proper. There is an analogous statement for cocycles.
Uniformly contracting means that there exists strictly distance contracting equivariant map . The infinitesimal version deals with a pair of an action and a cocycle, and assumes existence of an equivariant vectorfield which contracts in the following sense: the flow contracts.
Here is a short proof. Map an isometry to the unique fixed point of . This is continuous and equivariant. Properness at target implies properness at source.
9.3. Other orthogonal groups
One needs all orthogonal groups and their actions on spaces .
Theorem 21 Start with an isometric action of on space , preserving a proper convex domain and proper on it. Let be another action which preserves a proper convex domain and which is uniformly contracting in space-like directions with respect to . Then the pair is properly discontinuous.
There is an analogous statement for cocycles
The short argument does not generalize. Here is another proof which generalizes. I explain it in case . Fix and stick to its orbit. Map to the set of points in which minimize displacement by . It is a finite set. This equivariant map maps compact sets to compact sets, properness at target implies properness at source.
In the general case, convex sets are there to make sure that distances are used only in space-like directions.
9.4. Right-angled Coxeter groups
A finite graph, encoding commuting involutions, is given. The corresponding Gram matrix defines a symmetric bilinear form on , whence a canonical orthogonal representation of (Tits).
Let . This is nondegenerate of constant signature if , whence representations in . In , the -orbit of the fundamental polyhedron is properly convex, and is cocompact on it (Tits-Vinberg). We check that is uniformly contracting with respect to if . Also, the -derivative satisfies uniform contraction in cocycle sense.
need not be contained in . If so, replace it with the intersection with the dual convex set, which is contained in , and non-empty. Examples show that contraction in non space-like directions does not hold.
10. Olivier Guichard: Symplectic maximal representations
Joint work with Anna Wienhard.
For such representations, which are Anosov, we construct domains of discontinuity in projective spaces, and would like to understand the topology of quotients.
10.1. Maximal representations of surface groups
has a cyclic central extension , hence a characteristic class called the Euler number. On a surface subgroup equipped with the standard presentation, the Euler number is given by
This readily implies the Milnor-Wood inequality
Definition 22 Say is maximal if .
Burger-Iozzi-Labourie-Wienhard show that maximal representations are Anosov. The boundary embedding ,
is continuous, equivariant, antipodal (distinct points are mapped to transverse Lagrangians). It has a nice dynamical property: for every sequence in such that there exist distinct boundary points and such that, away from , converges to , then converge to away from the Lagrangians which are transverse to . This readily leads to proof of the proper discontinuity part of the following
Theorem 23 Define
If is Anosov, then the action of on the complement of is properly discontinuous and cocompact.
10.2. Proof of cocompactness I: dynamics
Inspired by Sullivan, Kapovitch-Leeb-Porti. We use their expansion result. For every , there exists , a neighborhood of in such that for all such that belongs to , for all ,
This implies cocompactness.
10.3. Proof of cocompactness II: cohomology
Easier if , since then is connected. We show that compactly supported cohomology . Integrating along fibers, this is equal to , which fits in the long exact sequence of the pair
So is a cokernel whose dimension does not depend on and can be computed for , it is equal to 1.
10.4. Proof of cocompactness III: maximality
A pair of transverse Lagrangians and defines a quadratic form on .
Definition 24 A triple of pairwise transverse Lagrangians is maximal of the the restriction to of the quadratic form associated to
Theorem 25 (Burger-Iozzi-Wienhard) is maximal iff the boundary embedding is maximal, i.e. for every oriented triple in , is maximal.
maps the oriented triple space of to the maximal triple space of the Lagrangian Grassmannian. This space carries a principal bundle , whose total space is the group itself. Let be the pulled-back bundle on triples. One shows that the map at the level of total spaces is open. It is equivariant. The -quotient is compact, therefore the image is open and compact. If , is connected, therefore this map is onto, and is compact.
10.5. Topology of quotients
Theorem 26 The quotient manifold fibres over with fibres homeomorphic to .
Since this statement is stable under deformation of the representation. We use knowledge of the connected components of the moduli space of maximal representations. Here is a list containing representatives of all connected components.
- Fuchsian representations .
- Fuchsian composed with the irreducible representation .
- Fuchsian representations composed with the fully reducible representation .
- A Fuchsian and an orthogonal representation composed with the tensor product representation .
- Amalgamation: glue two surfaces with boundary and along their boundary. Then is an amalgamation, compatible maximal representations define a representation of which is still maximal (Burger-Iozzi-Wienhard).
11. Sourav Ghosh: Moduli space of Margulis space-times
Margulis space times are examples of complete affine 3-manifolds with free fundamental groups. Their holonomy is contained in . The linear parts are discrete (Margulis started with certain Schottky groups).
In 1991, Drumm showed that one could start with an arbitrary Schottky group. He constructed fundamental domains bounded by what he called crooked planes. The fundamental domain of the Schottky group in hyperbolic plane is bounded by 4 lines. Add rays tangent to the isotropic circle. Consider the positive cone on this picture. Continue the picture to get complete half-planes. This bounds a fundamental domain.
Theorem 27 (Danciger-Gueritaud-Kassel) Any Margulis space time admits a fundamental domain bounded by crooked planes.
There are examples where the linear holonomy contains parabolics. From now on, I will stick to linear holonomies which are Schottky.
11.1. The neutralised section
The linear holonomy defines a hyperbolic surface . In its unit tangent bundle, let denote the set of bi-recurrent points of the geodesic flow, and its lift in the unit tangent bundle of hyperbolic plane. Similarly, there is a bi-recurrent set in the unit space-like tangent bundle of the affine 3-manifold .
Theorem 28 (Goldman-Labourie-Margulis) There exists a continuous equivariant map , called the neutralised section, such that
is an orbit preserving homeomorphism.
However, horocycles are not mapped to horocycles. Each of them is mapped into a fixed affine plane.
Theorem 29 Let us define the “new horocycles in ” as …. Then these sets are stable leafs under the geodesic flow of a metric on which is bi-Lipschitz equivalent to a Euclidean metric.
11.2. Affine Anosov representation
Definition 30 Let be a free group, let be two transverse null affine planes (parallel to planes tangent to the light cone). Let be their stabilizers in the affine group . We say that a representation of in is -affine Anosov if there is a continuous equivariant boundary map on pairs of distinct points of to , and a continuous equivariant map from the space of the geodesic flow of to the bundle of Euclidean metrics on showing contraction under the geodesic flow of .
Theorem 31 An affine action of a Schottky subgroup of on gives rise to a Margulis space time if and only if it is affine Anosov.
In higher dimensions, it is easier to establish the Anosov character of affine actions than to prove their properness.
12. Swiatoslav Gal: Simplicity of groups of dynamical origin
Certain simple groups are simpler than others.
Note that a group is simple iff for every nontrivial conjugacy class , the union of its powers is the whole group.
Say is boundedly simple if for every nontrivial conjugacy class , there exists such that
Full topological groups. Matsui proved that the commutator group of a full topological group is simple. His proof shows that it is boundedly simple. The exponent indeed depends on the element.
Say is uniformly simple if there exists such that for every nontrivial conjugacy class ,
A function is called a pseudo-length if and for all ,
Say is central if furthermore .
I observe that is simple iff every central pseudo-length is a length. Also, is boundedly simple iff every central pseudo-length is a bounded length. For uniform simplicity, I have only one implication,
uniformly simple every central pseudo-length is a bounded and discrete length.
12.2. Displacement and uniform simplicity
Burago-Ivanov-Polterovich discuss bi-invariant metrics on homeomorphism, diffeomorphism, symplectomorphism groups. They make the following observation: say that an element -displaces a subgroup if successive conjugated subgroups mutually commute. It this holds, then every element of the commutator subgroup with commutator length in fact ca be written as a prduct of two commutators,
The trick is an identity among commutators.
A consequence: if has the property that every finitely generated subgroup is -displaced by some conjugate of , then . In other words, is uniformly 6-simple.
12.3. Application: actions on ordered sets
Let act on a totally ordered set. Assume that
- action is bounded (the support of every group element is contained in some interval),
- action is proximal (every interval is mapped into any other interval by some group element).
Then the the commutator subgroup is uniformly 6-simple.
Note that bounded+proximal is equivalent to bounded+primitive (action is primitive if there are no factor ordered sets).
Example: Thompson’s group acting on the interval.
Example: affine action of on is proximal, not 2-transitive (which is an easy sufficient condition for proximality) but unbounded.
12.4. Groups of PL maps of the interval
Here are examples of groups of interest. Automorphism group of a tree. The corresponding full topological group (known as Neretin group). Group of planar tree automorphisms (for some planar embedding of the tree) and its full topological group.
Theorem 32 Equivalent properties for a group acting on the ideal boundary of a tree.
- is proximal.
- The full group is proximal.
- action is minimal and does not preserve any probability measure.
- action is minimal and non-parabolic.
Theorem 33 If and is proximal, then is uniformly simple.
12.5. More groups
The quasi-isometry group of a tree is uniformly simple as well (with Nir Lazarovitsch). The bound is better than in the previous class.
Volodymyr Nekrashevich has a paper where he shows that certain full topological groups are simple and finitely generated. He uses a different language.