12 Kleinian groups
Problem 12.1 (Comments) [Misha Kapovich] For the fundamental group
dvipng error! exitcode was 2 (signal 0), transscript follows:of a closed hyperbolic
dvipng error! exitcode was 2 (signal 0), transscript follows:-manifold consider a short exact sequence
dvipng error! exitcode was 2 (signal 0), transscript follows:
Is the group
dvipng error! exitcode was 2 (signal 0), transscript follows:residually finite? In other words, is there a finite-index subgroup
dvipng error! exitcode was 2 (signal 0), transscript follows:in
dvipng error! exitcode was 2 (signal 0), transscript follows:so that the restriction map
dvipng error! exitcode was 2 (signal 0), transscript follows:
is zero? Remarkably, positive answer is presently known only for
dvipng error! exitcode was 2 (signal 0), transscript follows:. Same problem makes sense also for the fundamental groups of complex-hyperbolic and quaternionic-hyperbolic manifolds.
Problem 12.2 (Comments) [Misha Kapovich] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be as above. Is there a finite-index subgroup
dvipng error! exitcode was 2 (signal 0), transscript follows:so that the restriction map
dvipng error! exitcode was 2 (signal 0), transscript follows:
is zero?
This problem is interesting because
dvipng error! exitcode was 2 (signal 0), transscript follows:classifies PL structures on the hyperbolic manifold
dvipng error! exitcode was 2 (signal 0), transscript follows:, see MR0645390.
Problem 12.3 (Comments) [Misha Kapovich] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a Gromov-hyperbolic Coxeter group. Does
dvipng error! exitcode was 2 (signal 0), transscript follows:admit a discrete embedding in
dvipng error! exitcode was 2 (signal 0), transscript follows:for large
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Note that the Coxeter generators are not assumed to act as reflections on
dvipng error! exitcode was 2 (signal 0), transscript follows:. Otherwise, there are counter-examples, see Felikson-Tumarkin.
Problem 12.4 (Comments) (Misha Kapovich) Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a convex-cocompact subgroup of isometries of complex-hyperbolic 2-space. Can the limit set of
dvipng error! exitcode was 2 (signal 0), transscript follows:be homeomorphic to the Sierpinski carpet?
Problem 12.5 (Comments) [Misha Kapovich] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a discrete torsion-free finitely-generated subgroup without abelian subgroups of rank
dvipng error! exitcode was 2 (signal 0), transscript follows:. Is it true that
(a)
dvipng error! exitcode was 2 (signal 0), transscript follows:
Here
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:is the round sphere and
dvipng error! exitcode was 2 (signal 0), transscript follows:? This is known to be true in the case when
dvipng error! exitcode was 2 (signal 0), transscript follows:is geometrically finite Kapovich(2007).
(c) If
dvipng error! exitcode was 2 (signal 0), transscript follows:, is it true that
dvipng error! exitcode was 2 (signal 0), transscript follows:is geometrically finite?
(d) If
dvipng error! exitcode was 2 (signal 0), transscript follows:, does it follow that
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:be a Schottky group (or, more generally, a free convex-cocompact group). Can Hausdorff dimension of the limit set of
dvipng error! exitcode was 2 (signal 0), transscript follows:be arbitrarily close to
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Problem 12.7 (Comments) [Misha Kapovich] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a finitely-generated discrete group of isometries of a Gromov-hyperbolic space
dvipng error! exitcode was 2 (signal 0), transscript follows:so that the limit set of
dvipng error! exitcode was 2 (signal 0), transscript follows:is connected. Is it true that the limit set of
dvipng error! exitcode was 2 (signal 0), transscript follows:is locally connected?
Consider a representation
dvipng error! exitcode was 2 (signal 0), transscript follows:. This action of
dvipng error! exitcode was 2 (signal 0), transscript follows:on the hyperbolic space determines a class function
dvipng error! exitcode was 2 (signal 0), transscript follows:
so that
dvipng error! exitcode was 2 (signal 0), transscript follows:is the displacement for the isometry
dvipng error! exitcode was 2 (signal 0), transscript follows:of
dvipng error! exitcode was 2 (signal 0), transscript follows:, i.e.,
dvipng error! exitcode was 2 (signal 0), transscript follows:
Problem 12.8 (Comments) Suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:are discrete and faithful representations as above so that there exists
dvipng error! exitcode was 2 (signal 0), transscript follows:for which we have
dvipng error! exitcode was 2 (signal 0), transscript follows:
Does it follow that there exists a quasiconformal map
dvipng error! exitcode was 2 (signal 0), transscript follows:which is equivariant with respect to the isomorphism
dvipng error! exitcode was 2 (signal 0), transscript follows:? Can one choose
dvipng error! exitcode was 2 (signal 0), transscript follows:which is
dvipng error! exitcode was 2 (signal 0), transscript follows:-quasiconformal for
dvipng error! exitcode was 2 (signal 0), transscript follows:?
If
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:be a finitely-presented group which does not split as a graph of groups with virtually abelian edge groups. For every
dvipng error! exitcode was 2 (signal 0), transscript follows:define the space
dvipng error! exitcode was 2 (signal 0), transscript follows:
of conjugacy classes of discrete and faithful representations of
dvipng error! exitcode was 2 (signal 0), transscript follows:into
dvipng error! exitcode was 2 (signal 0), transscript follows:. We assume that
dvipng error! exitcode was 2 (signal 0), transscript follows:is not virtually abelian itself. Then Rips' theory of group actions on trees implies that
dvipng error! exitcode was 2 (signal 0), transscript follows:is compact.
Problem 12.9 (Comments) \label{constructive} Find a "constructive" proof of the above theorem. More precisely, consider a finite presentation
dvipng error! exitcode was 2 (signal 0), transscript follows:of
dvipng error! exitcode was 2 (signal 0), transscript follows:. Given
dvipng error! exitcode was 2 (signal 0), transscript follows:define
dvipng error! exitcode was 2 (signal 0), transscript follows:
Find an explicit constant
dvipng error! exitcode was 2 (signal 0), transscript follows:, which depends on
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:and the lengths of the words
dvipng error! exitcode was 2 (signal 0), transscript follows:, so that the function
dvipng error! exitcode was 2 (signal 0), transscript follows:is bounded from above by
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Remark Y.~Lai Lai found such an explicit constant
dvipng error! exitcode was 2 (signal 0), transscript follows:can be for Coxeter groups; moreover, in this case
dvipng error! exitcode was 2 (signal 0), transscript follows:depends only on
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:for some
dvipng error! exitcode was 2 (signal 0), transscript follows:.
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
dvipng error! exitcode was 2 (signal 0), transscript follows:
is the fundamental group of a compact K\"ahler manifold, then every homomorphismdvipng error! exitcode was 2 (signal 0), transscript follows:
wither factors through a group commensurable to a surface group ordvipng error! exitcode was 2 (signal 0), transscript follows:
preserves a point or a pair of points indvipng error! exitcode was 2 (signal 0), transscript follows:
. See MR1019964.
Problem 12.10 (Comments) Find new restrictions on Kleinian groups.
Recall that a group
dvipng error! exitcode was 2 (signal 0), transscript follows:is called coherent if every finitely-generated subgroup of
dvipng error! exitcode was 2 (signal 0), transscript follows:is finitely-presented.
Problem 12.11 (Comments) [M.~Kapovich, L.~Potyagailo, E.B.~Vinberg] Prove that every arithmetic lattice in
dvipng error! exitcode was 2 (signal 0), transscript follows:(
dvipng error! exitcode was 2 (signal 0), transscript follows:) is non-coherent.
See KPV for some partial results in this direction.
It is well-known that every lattice in
dvipng error! exitcode was 2 (signal 0), transscript follows:(
dvipng error! exitcode was 2 (signal 0), transscript follows:) has Property T.
Problem 12.12 (Comments) Suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:is a discrete subgroup satisfying Property T. Does it follow that
dvipng error! exitcode was 2 (signal 0), transscript follows:preserves a totally-geodesic subspace
dvipng error! exitcode was 2 (signal 0), transscript follows:in
dvipng error! exitcode was 2 (signal 0), transscript follows:and acts on
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:. Does it follow that
dvipng error! exitcode was 2 (signal 0), transscript follows:has Property T?
Problem 12.14 (Comments) Generalize Bestvina-Feighn combination theorem from graphs of gro\-ups to complexes of groups.
Background: Let
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:is an isometry of finite order which fixes a (complex) codimension 1 hyperplane. A reflection group in
dvipng error! exitcode was 2 (signal 0), transscript follows:is a subgroup of
dvipng error! exitcode was 2 (signal 0), transscript follows:generated by reflections. These concepts generalize the notion of reflections and reflection groups acting on
dvipng error! exitcode was 2 (signal 0), transscript follows:. Vinberg MR774946 proved that there for
dvipng error! exitcode was 2 (signal 0), transscript follows:there are no uniform lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:which are reflection groups. This result was extended by Prokhorov MR842588 who proved nonexistence of reflection lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:for
dvipng error! exitcode was 2 (signal 0), transscript follows:.
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
dvipng error! exitcode was 2 (signal 0), transscript follows:such that for
dvipng error! exitcode was 2 (signal 0), transscript follows:, there are no lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:.
A better definition replacing convex-cocompactness could be:
A finitely-generated group
dvipng error! exitcode was 2 (signal 0), transscript follows:is undistorted if the induced map from the Cayley graph of
dvipng error! exitcode was 2 (signal 0), transscript follows:to
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:, 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
dvipng error! exitcode was 2 (signal 0), transscript follows:, where
dvipng error! exitcode was 2 (signal 0), transscript follows:is a semisimple Lie group. In the case when
dvipng error! exitcode was 2 (signal 0), transscript follows:is a surface group and
dvipng error! exitcode was 2 (signal 0), transscript follows:, this notion can be reformulated in terms of action of
dvipng error! exitcode was 2 (signal 0), transscript follows:on its limit set in
dvipng error! exitcode was 2 (signal 0), transscript follows:, i.e. existence of a
dvipng error! exitcode was 2 (signal 0), transscript follows:-invariant hyperconvex curve in
dvipng error! exitcode was 2 (signal 0), transscript follows:.
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
dvipng error! exitcode was 2 (signal 0), transscript follows:in order to prove rigidity results for embeddings of lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:into other higher rank Lie groups.
Background. Ideal boundaries of totally-geodesic subspaces
dvipng error! exitcode was 2 (signal 0), transscript follows:define holomorphic chains in
dvipng error! exitcode was 2 (signal 0), transscript follows:. 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
dvipng error! exitcode was 2 (signal 0), transscript follows:of maximal (symplectic) area. The incidence relation between holomorphic chains in
dvipng error! exitcode was 2 (signal 0), transscript follows:determines a "building-like" structure where chains serve as apartments: Every two points belong to a chain. Given a measurable map
dvipng error! exitcode was 2 (signal 0), transscript follows:
which induces a measurable morphism of these "building-like" structures, is induced by a holomorphic embedding
dvipng error! exitcode was 2 (signal 0), transscript follows:. This, in turn, can be used to reprove Corlette's rigidity theorem MR965220 for representations of lattices in
dvipng error! exitcode was 2 (signal 0), transscript follows:into
dvipng error! exitcode was 2 (signal 0), transscript follows:. The motivation for the Problem \ref{w2} is to extend Corlette's rigidity result to representations of
dvipng error! exitcode was 2 (signal 0), transscript follows: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.