Boundaries/Section12 - Problems in Geometric Group Theory

12 Kleinian groups

Problem 12.1 (Comments) [Misha Kapovich] For the fundamental group $G$ of a closed hyperbolic $n$ -manifold consider a short exact sequence

$1\to \Z_p \to \Gamma\to G\to 1.$

Is the group $\Gamma$ residually finite? In other words, is there a finite-index subgroup $G'$ in $G$ so that the restriction map

$H^2(G, \Z_p)\to H^2(G', \Z_p)$

is zero? Remarkably, positive answer is presently known only for $n=2$ . Same problem makes sense also for the fundamental groups of complex-hyperbolic and quaternionic-hyperbolic manifolds.

Problem 12.2 (Comments) [Misha Kapovich] Let $G$ be as above. Is there a finite-index subgroup $G'\subset G$ so that the restriction map

$H^3(G, \Z_2)\to H^3(G', \Z_2)$

is zero?

This problem is interesting because $H^3(G, \Z_2)$ classifies PL structures on the hyperbolic manifold $\H^n/G$ , see MR0645390.

Problem 12.3 (Comments) [Misha Kapovich] Let $G$ be a Gromov-hyperbolic Coxeter group. Does $G$ admit a discrete embedding in $Isom(\H^n)$ for large $n$ ?

Note that the Coxeter generators are not assumed to act as reflections on $\H^n$ . Otherwise, there are counter-examples, see Felikson-Tumarkin.

Problem 12.4 (Comments) (Misha Kapovich) Let $G\subset PU(2,1)$ be a convex-cocompact subgroup of isometries of complex-hyperbolic 2-space. Can the limit set of $G$ be homeomorphic to the Sierpinski carpet?

Problem 12.5 (Comments) [Misha Kapovich] Let $G\subset Isom(\H^n)$ be a discrete torsion-free finitely-generated subgroup without abelian subgroups of rank $\ge 2$ . Is it true that

(a)

$cd_\Z(G)\le Hdim(\La_c(G))+1 \quad ?$

Here $\La_c$ is the conical limit set. The answer is known Kapovich(2007) to be positive if one considers homological rather than cohomological dimension.

(b) In the case of equality, is it true that the limit set of $G$ is the round sphere and $G$ ? This is known to be true in the case when $G$ is geometrically finite Kapovich(2007).

(d) If $Hdim(\La_c(G))<1$ , does it follow that $G$ is a classical Schottky-type group? (I.e. the one whose fundamental domain is bounded by round spheres.) See Hou for partial results.

Problem 12.6 (Comments) [Lewis Bowen] Let $G\subset Isom(\H^4)$ be a Schottky group (or, more generally, a free convex-cocompact group). Can Hausdorff dimension of the limit set of $G$ be arbitrarily close to $3$ ?

Problem 12.7 (Comments) [Misha Kapovich] Let $G$ be a finitely-generated discrete group of isometries of a Gromov-hyperbolic space $X$ so that the limit set of $G$ is connected. Is it true that the limit set of $G$ is locally connected?

Consider a representation $\rho: G\to \Isom(\H^n)$ . This action of $G$ on the hyperbolic space determines a class function

$\ell_\rho: G\to \R_+,$

so that $\ell_\rho(g)$ is the displacement for the isometry $\rho(g)$ of $\H^n$ , i.e.,

$\ell_\rho(g)=\inf_{x\in \H^n} d(\rho(g)(x), x).$

Problem 12.8 (Comments) Suppose that $\rho_1, \rho_2$ are discrete and faithful representations as above so that there exists $C>0$ for which we have

$C^{-1} \le \frac{\ell_{\rho_1}(g)}{\ell_{\rho_2}(g)})\le C, \quad\forall g\in G.$

Does it follow that there exists a quasiconformal map $f:\La(\rho_1(G))\to \La(\rho_2(G))$ which is equivariant with respect to the isomorphism $\rho_2\circ \rho_1^{-1}$ ? Can one choose $f$ which is $K$ -quasiconformal for $K=K(C)$ ?

If $n=3$ and $G$ is finitely generated, then the answer to the first part of the problem is positive and follows from the solution of the ending lamination conjecture.

A constructive proof of Rips compactness theorem. Let $G$ be a finitely-presented group which does not split as a graph of groups with virtually abelian edge groups. For every $n$ define the space

${\mathcal D}_n(G)$

of conjugacy classes of discrete and faithful representations of $G$ into $\Isom(\H^n)$ . We assume that $G$ is not virtually abelian itself. Then Rips' theory of group actions on trees implies that ${\mathcal D}_n(G)$ is compact.

Problem 12.9 (Comments) \label{constructive} Find a "constructive" proof of the above theorem. More precisely, consider a finite presentation $\<g_1,..,g_k|R_1,..,R_m\>$ of $G$ . Given $[\rho]\in {\mathcal D}_{n}(G)$ define

$B_{n}([\rho]):= \inf_{x\in \H^{n}} \max_{i=1,...,k} d(x,\rho(g_i)(x)).$

Find an explicit constant $C$ , which depends on $n$ , $k, m$ and the lengths of the words $R_i$ , so that the function $B_{n}:{\mathcal D}_{n}(G)\to \R$ is bounded from above by $C$ .

Remark Y.~Lai Lai found such an explicit constant $C$ can be for Coxeter groups; moreover, in this case $C$ depends only on $n$ and the number of Coxeter generators.

One possible application of the solution of Problem \ref{constructive} is in producing nontrivial algebraic restrictions on Kleinian groups.

An abstract Kleinian group is a group which admits a discrete embedding in $\Isom(\H^n)$ for some $n$ .

All currently known algebraic restrictions on finitely-generated Kleinian groups can be traced to the following:

Every Kleinian group has the Haagerup property: They admit isometric properly discontinuous actions on some Hilbert space. See for instance MR1852148.
If $\pi$ is the fundamental group of a compact K\"ahler manifold, then every homomorphism $\rho:\pi\to \Isom(\H^n)$ wither factors through a group commensurable to a surface group or $\rho(\pi)$ preserves a point or a pair of points in $\H^n\cup \geo \H^n$ . See MR1019964.

Problem 12.10 (Comments) Find new restrictions on Kleinian groups.

Recall that a group $G$ is called coherent if every finitely-generated subgroup of $G$ is finitely-presented.

Problem 12.11 (Comments) [M.~Kapovich, L.~Potyagailo, E.B.~Vinberg] Prove that every arithmetic lattice in $\Isom(\H^n)$ ( $n\ge 4$ ) is non-coherent.

See KPV for some partial results in this direction.

It is well-known that every lattice in $\Isom(\H \H^n)$ ( $n\ge 2$ ) has Property T.

Problem 12.12 (Comments) Suppose that $G\subset \Isom(\H \H^n)$ is a discrete subgroup satisfying Property T. Does it follow that $G$ preserves a totally-geodesic subspace $H$ in $\H \H^n$ and acts on $H$ as a lattice?

The main motivation for this problem comes from the fact that the obvious constrictions of discrete groups of isometries are by various graphs of groups and hence these groups do not have Property T. One can try to use triangles of groups:

Problem 12.13 (Comments) Suppose that $\Delta$ is a developable triangle of groups, where all the cell-groups have Property T and so that all the links in the universal cover of T have $\la_1>1/2$ . Does it follow that $\pi_1(\Delta)$ has Property T?

Problem 12.14 (Comments) Generalize Bestvina-Feighn combination theorem from graphs of gro\-ups to complexes of groups.

Background: Let ${\mathcal G}$ be a graph of groups, so that vertex and edge groups are hyperbolic and the edge subgroup are quasiconvex in the vertex groups. Bestvina and Feighn MR1152226 found some sufficient conditions for $\pi_1({\mathcal G})$ to be hyperbolic. Hammenst\"adt has some partial results towards solving this problem.

Discrete subgroups in other Lie groups.

A reflection in a complex-hyperbolic space $\C \H^n$ is an isometry of finite order which fixes a (complex) codimension 1 hyperplane. A reflection group in $\C \H^n$ is a subgroup of $PU(n,1)$ generated by reflections. These concepts generalize the notion of reflections and reflection groups acting on $\H^n$ . Vinberg MR774946 proved that there for $n\ge 30$ there are no uniform lattices in $O(n,1)$ which are reflection groups. This result was extended by Prokhorov MR842588 who proved nonexistence of reflection lattices in for $n\ge 996$ .

Problem 12.15 (Comments) Generalize Vinberg's finiteness theorem for reflection groups to complex-hyperbolic reflection groups, i.e., prove that there exists a number $N$ such that for $n\ge N$ , there are no lattices in $PU(n,1)$ which are generated by reflections.

Problem 12.16 (Comments) [Misha Kapovich] There is a theory of quasi-convex groups acting on Gromov hyperbolic spaces, generalizing the theory of convex-compact groups of isometries of the real hyperbolic space. Develop a theory of geometric finiteness in CAT(0) spaces.

Remark \label{convexrare} It is a priori unclear what to take as the definition of geometric finiteness in the context of CAT(0) spaces (even in the case of symmetric spaces). Taking quotients of the convex hull is a bad idea, as shown by a theorem of Bruce Kleiner and Bernhard Leeb: There are only few convex subsets in symmetric spaces of rank $\ge 2$ .

A better definition replacing convex-cocompactness could be:

A finitely-generated group $G\subset {\rm Isom}(X)$ is undistorted if the induced map from the Cayley graph of $G$ to $X$ is a quasi-isometric embedding.

In the case of Gromov hyperbolic spaces, undistorted is equivalent to quasi-convex.

There are examples of undistorted free Zariski dense subgroups of $SL(n,\R)$ , generalizing the Schottky construction.

Is there an interpretation of the notion of undistorted groups in terms of the group actions on limit sets?

F.~Labourie MR2221137 introduced another notion of convex-cocompactness that he calls an Anosov structure, for group representations $\rho: \Gamma\to G$ , where $G$ is a semisimple Lie group. In the case when $\Gamma$ is a surface group and $G=SL(n+1,\R)$ , this notion can be reformulated in terms of action of $\rho(\Gamma)$ on its limit set in $\R P^n$ , i.e. existence of a $\rho(\Gamma)$ -invariant hyperconvex curve in $\R P^n$ .

Problem 12.17 (Comments) [Anna Wienhard] Extend this relation of Anosov structure and dynamics on the limit set to representations of other hyperbolic groups.

Problem 12.18 (Comments) [Anna Wienhard]\label{w2} Generalize holomorphic chain patterns in $\geo \C \H^n$ in order to prove rigidity results for embeddings of lattices in $PU(n,1)$ into other higher rank Lie groups.

Background. Ideal boundaries of totally-geodesic subspaces $\C\H^1\subset \C H^n$ define holomorphic chains in $\geo \C \H^n$ . These circles are characterized by the property that three points belong to such a chain if and only if they span an ideal triangle in $\C \H^n$ of maximal (symplectic) area. The incidence relation between holomorphic chains in $\geo \C \H^n$ determines a "building-like" structure where chains serve as apartments: Every two points belong to a chain. Given a measurable map

$\geo \C\H^n\to \geo \C \H^m, m\ge n,$

which induces a measurable morphism of these "building-like" structures, is induced by a holomorphic embedding $\C\H^n\to \C\H^m$ . This, in turn, can be used to reprove Corlette's rigidity theorem MR965220 for representations of lattices in $PU(n,1)$ into $PU(m,1)$ . The motivation for the Problem \ref{w2} is to extend Corlette's rigidity result to representations of to other Lie groups.

Problem 12.19 (Comments) [Anna Wienhard] Obtain new rigidity results for embeddings of real-hyperbolic lattices into higher-rank semisimple Lie groups in terms of the boundary maps.

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