13 Miscellaneous problems in geometric group theory
Problem 13.1 (Comments) [Kevin Whyte: Homotopy Nielsen realization] If
dvipng error! exitcode was 2 (signal 0), transscript follows:is a compact polyhedron and
dvipng error! exitcode was 2 (signal 0), transscript follows:is a discrete group of simple homotopy equivalences
dvipng error! exitcode was 2 (signal 0), transscript follows:, is there a compact space
dvipng error! exitcode was 2 (signal 0), transscript follows:, homotopy equivalent to
dvipng error! exitcode was 2 (signal 0), transscript follows:, such that
dvipng error! exitcode was 2 (signal 0), transscript follows:can be realized as a group of homeomorphisms of
dvipng error! exitcode was 2 (signal 0), transscript follows:. \label{hnr}
Remark It is a long-standing open problem to determine if the exact sequence
dvipng error! exitcode was 2 (signal 0), transscript follows:
is split. Here
dvipng error! exitcode was 2 (signal 0), transscript follows:is a compact surface of genus
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:denotes the connected component of the identity in the group of homeomorphisms.
Moreover, there are examples due to George Cooke of spaces
dvipng error! exitcode was 2 (signal 0), transscript follows:and finite groups
dvipng error! exitcode was 2 (signal 0), transscript follows:of simple homotopy-equivalences of
dvipng error! exitcode was 2 (signal 0), transscript follows:for which the answer to Problem~\ref{hnr} is "No". However all known examples occur in dimensions
dvipng error! exitcode was 2 (signal 0), transscript follows:and require the group to have torsion.
Problem 13.2 (Comments) [Ilia Kapovich] Consider finite cell complexes
dvipng error! exitcode was 2 (signal 0), transscript follows:. Is there an algorithm to determine if
dvipng error! exitcode was 2 (signal 0), transscript follows:is contractible?
Remark The triviality problem is known to be unsolvable for finitely presented groups. However, their presentation complexes are never contractible, since the presentation is unbalanced. On the other hand, for complexes of dimension
dvipng error! exitcode was 2 (signal 0), transscript follows:, there is no algorithm to determine contractibility (S.~Weinberger MR1308714). Indeed, take a triangulated closed
dvipng error! exitcode was 2 (signal 0), transscript follows:-manifold
dvipng error! exitcode was 2 (signal 0), transscript follows:for which it is impossible to decide if
dvipng error! exitcode was 2 (signal 0), transscript follows:is homeomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be the complement to an open
dvipng error! exitcode was 2 (signal 0), transscript follows:-simplex in
dvipng error! exitcode was 2 (signal 0), transscript follows:. Then contractibility of
dvipng error! exitcode was 2 (signal 0), transscript follows:is undecidable, since it is equivalent to
dvipng error! exitcode was 2 (signal 0), transscript follows:being a homotopy sphere.
Remark [Daniel Groves] It is an open problem if the triviality of a group is algorithmically solvable for groups with balanced presentation. (A presentation is called balanced if the number of generators equals the number of relators.)
Problem 13.3 (Comments) [Kevin Whyte] For a word-hyperbolic
dvipng error! exitcode was 2 (signal 0), transscript follows:not splitting over any virtually cyclic group, can an infinite-index subgroup and a finite-index subgroup be isomorphic?
Remark This asks for something slightly stronger than the cohopfian property.
Problem 13.4 (Comments) [Misha Kapovich] Consider Teichm\"uller space
dvipng error! exitcode was 2 (signal 0), transscript follows:with Teichm\"uller metric. Does it have quadratic isoperimetric inequality?
Background: If
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:is known to be non-hyperbolic. However the Mapping Class Group is bi-automatic, therefore the "thick part" of
dvipng error! exitcode was 2 (signal 0), transscript follows:is semihyperbolic. One can ask a similar question for the outer space.
Curt McMullen defined "inflexibility" for Kleinian groups, MR1401347.
Problem 13.5 (Comments) [Danny Calegari] Is there a similar statement to this inflexibility result this with no group specified---that is, for subsets
dvipng error! exitcode was 2 (signal 0), transscript follows:of the boundary sphere of
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Here is a possible setup for such problem: Define a random Beltrami differential as follows. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be the tessellation of
dvipng error! exitcode was 2 (signal 0), transscript follows:by regular right-angled hyperbolic pentagons. (All such pentagons are isometric to a model pentagon
dvipng error! exitcode was 2 (signal 0), transscript follows:.) Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a compact (perhaps finite or even a singleton) set of Beltrami differentials on
dvipng error! exitcode was 2 (signal 0), transscript follows:having norm
dvipng error! exitcode was 2 (signal 0), transscript follows:(or any fixed number
dvipng error! exitcode was 2 (signal 0), transscript follows:). For concreteness, suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:is a singleton. For each pentagon
dvipng error! exitcode was 2 (signal 0), transscript follows:choose a random isometry
dvipng error! exitcode was 2 (signal 0), transscript follows:. (There are 10 such isometries.) Then push-forward
dvipng error! exitcode was 2 (signal 0), transscript follows:from
dvipng error! exitcode was 2 (signal 0), transscript follows:to
dvipng error! exitcode was 2 (signal 0), transscript follows:via
dvipng error! exitcode was 2 (signal 0), transscript follows:. This defines a \em{ random Beltrami differential}
dvipng error! exitcode was 2 (signal 0), transscript follows:on
dvipng error! exitcode was 2 (signal 0), transscript follows:. Given a closed connected set
dvipng error! exitcode was 2 (signal 0), transscript follows:observe that each complementary component
dvipng error! exitcode was 2 (signal 0), transscript follows:is simply-connected. Choose a Riemann mapping
dvipng error! exitcode was 2 (signal 0), transscript follows:; push-forward the random Beltrami differential
dvipng error! exitcode was 2 (signal 0), transscript follows:from
dvipng error! exitcode was 2 (signal 0), transscript follows:to
dvipng error! exitcode was 2 (signal 0), transscript follows:. Repeat this for each component of
dvipng error! exitcode was 2 (signal 0), transscript follows:and extend the resulting differential to
dvipng error! exitcode was 2 (signal 0), transscript follows:by zero. This defines a Beltrami differential
dvipng error! exitcode was 2 (signal 0), transscript follows:on
dvipng error! exitcode was 2 (signal 0), transscript follows:. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be the quasiconformal map which is a solution of the Beltrami equation
dvipng error! exitcode was 2 (signal 0), transscript follows:
Such a quasiconformal map has a natural Thurston-Reimann biLipschitz extension
dvipng error! exitcode was 2 (signal 0), transscript follows:, MR0802511.
Problem 13.6 (Comments) Given
dvipng error! exitcode was 2 (signal 0), transscript follows:, estimate the biLipschitz constant of
dvipng error! exitcode was 2 (signal 0), transscript follows:near
dvipng error! exitcode was 2 (signal 0), transscript follows:in terms of the distance
dvipng error! exitcode was 2 (signal 0), transscript follows:from
dvipng error! exitcode was 2 (signal 0), transscript follows:to the exterior of the convex hull of
dvipng error! exitcode was 2 (signal 0), transscript follows:.
More concretely: if
dvipng error! exitcode was 2 (signal 0), transscript follows:is a quasicircle, is the decay exponential in
dvipng error! exitcode was 2 (signal 0), transscript follows:? That is, are there positive constants
dvipng error! exitcode was 2 (signal 0), transscript follows:such that
dvipng error! exitcode was 2 (signal 0), transscript follows:
where
dvipng error! exitcode was 2 (signal 0), transscript follows:is the bilipschitz constant of
dvipng error! exitcode was 2 (signal 0), transscript follows:restricted to the ball of some fixed radius (say radius 1) about
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:is the convex hull of
dvipng error! exitcode was 2 (signal 0), transscript follows:, and
dvipng error! exitcode was 2 (signal 0), transscript follows:is a point in the interior of
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 13.7 (Comments) [Mladen Bestvina] Are braid groups CAT(0)?
Remark It is conjectured that all Artin groups are CAT(0).
Problem 13.8 (Comments) Extend Rips' theory to higher-dimensional buildings, e.g. products of
dvipng error! exitcode was 2 (signal 0), transscript follows:-trees.
Rank rigidity. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a
dvipng error! exitcode was 2 (signal 0), transscript follows:metric space. The space
dvipng error! exitcode was 2 (signal 0), transscript follows:is said to be or rank
dvipng error! exitcode was 2 (signal 0), transscript follows:if every geodesic segment in
dvipng error! exitcode was 2 (signal 0), transscript follows:is contained in a subset
dvipng error! exitcode was 2 (signal 0), transscript follows:which is isometric to a flat
dvipng error! exitcode was 2 (signal 0), transscript follows:-dimensional parallelepiped. If
dvipng error! exitcode was 2 (signal 0), transscript follows:is a locally
dvipng error! exitcode was 2 (signal 0), transscript follows:metric space, then
dvipng error! exitcode was 2 (signal 0), transscript follows:is said to have rank
dvipng error! exitcode was 2 (signal 0), transscript follows:if its universal cover is of rank
dvipng error! exitcode was 2 (signal 0), transscript follows:. The rank rigidity theorem proven by Ballmann MR819559 and by Burns and Spatzier MR908215, MR908214 states that:
If
dvipng error! exitcode was 2 (signal 0), transscript follows:is a compact nonpositively curved Riemannian manifold of rank
dvipng error! exitcode was 2 (signal 0), transscript follows:, then either
dvipng error! exitcode was 2 (signal 0), transscript follows:admits a finite cover the universal cover of
dvipng error! exitcode was 2 (signal 0), transscript follows:splits (nontrivially) as a Riemannian direct product or
dvipng error! exitcode was 2 (signal 0), transscript follows:is a locally symmetric space.
Problem 13.9 (Comments) [Werner Ballmann, Misha Brin] Suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:is a compact finite-dimensional locally
dvipng error! exitcode was 2 (signal 0), transscript follows:metric space of rank
dvipng error! exitcode was 2 (signal 0), transscript follows:. Then either the universal cover of
dvipng error! exitcode was 2 (signal 0), transscript follows:splits (nontrivially) as a Riemannian direct product or it is isometric to a Euclidean building.
This problem is most natural in the context of piecewise-Euclidean metric cell complexes. The conjecture was proven in the case of 2-dimensional and 3-dimensional complexes by Ballmann and Brin MR1383216, MR1781923.
Cogrowth. Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a subgroup of a finitely-generated group
dvipng error! exitcode was 2 (signal 0), transscript follows:. The cogrowth of
dvipng error! exitcode was 2 (signal 0), transscript follows:in
dvipng error! exitcode was 2 (signal 0), transscript follows:is the growth of the Shreier graph
dvipng error! exitcode was 2 (signal 0), transscript follows:, where
dvipng error! exitcode was 2 (signal 0), transscript follows:is a Cayley graph of
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 13.10 (Comments) Compute cogrowth for "interesting" subgroups. For instance:
Show that the cogrowth of
dvipng error! exitcode was 2 (signal 0), transscript follows:
indvipng error! exitcode was 2 (signal 0), transscript follows:
is exponential.- Compute cogrowth of special subgroups in Coxeter groups. (See
Vish for partial results.)
Suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:
is Gromov-hyperbolic. Is it true that the cogrowth is either constant, linear or exponential?
Coarse Whitehead Conjecture.
Problem 13.11 (Comments) [Whitehead Conjecture] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be an aspherical (i.e. with contractible universal cover) 2-dimensional complex. Is it true that every subcomplex of
dvipng error! exitcode was 2 (signal 0), transscript follows:is also aspherical? (See MR0004123.)
A metric space
dvipng error! exitcode was 2 (signal 0), transscript follows:is said to be coarsely trivial
dvipng error! exitcode was 2 (signal 0), transscript follows:if the following holds: There exists a function
dvipng error! exitcode was 2 (signal 0), transscript follows:so that for each
dvipng error! exitcode was 2 (signal 0), transscript follows:the map
dvipng error! exitcode was 2 (signal 0), transscript follows:
induces zero map of the
dvipng error! exitcode was 2 (signal 0), transscript follows:-th homotopy groups. For instance, suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:is the 0-skeleton of an
dvipng error! exitcode was 2 (signal 0), transscript follows:-connected simplicial complex
dvipng error! exitcode was 2 (signal 0), transscript follows:, which admits a cocompact free group action. Metrize
dvipng error! exitcode was 2 (signal 0), transscript follows:by declaring each edge of
dvipng error! exitcode was 2 (signal 0), transscript follows:to have unit length. Then
dvipng error! exitcode was 2 (signal 0), transscript follows:has coarsely trivial
dvipng error! exitcode was 2 (signal 0), transscript follows:for all
dvipng error! exitcode was 2 (signal 0), transscript follows:. Given a 2-dimensional contractible complex
dvipng error! exitcode was 2 (signal 0), transscript follows:as above and a connected subgraph
dvipng error! exitcode was 2 (signal 0), transscript follows:, metrize
dvipng error! exitcode was 2 (signal 0), transscript follows:using the above path-metric on
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 13.12 (Comments) ["Coarse Whitehead Conjecture", Misha Kapovich] Under the above assumptions, is it true that
dvipng error! exitcode was 2 (signal 0), transscript follows:has coarsely trivial
dvipng error! exitcode was 2 (signal 0), transscript follows:for
dvipng error! exitcode was 2 (signal 0), transscript follows:?
More restrictively one can consider the case when
dvipng error! exitcode was 2 (signal 0), transscript follows:is the Cayley complex of a finitely-presented group
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:is finitely-generated subgroup of
dvipng error! exitcode was 2 (signal 0), transscript follows:, identified with
dvipng error! exitcode was 2 (signal 0), transscript follows:. (Then the metric on
dvipng error! exitcode was 2 (signal 0), transscript follows:is quasi-isometric to the word metric on
dvipng error! exitcode was 2 (signal 0), transscript follows:.) Note that if
dvipng error! exitcode was 2 (signal 0), transscript follows:is finitely-presented then (since its cohomological dimension
dvipng error! exitcode was 2 (signal 0), transscript follows:) it has finite type and, thus
dvipng error! exitcode was 2 (signal 0), transscript follows:necessarily has coarsely trivial
dvipng error! exitcode was 2 (signal 0), transscript follows:for all
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 13.13 (Comments) Does the Coarse Whitehead Conjecture hold if
dvipng error! exitcode was 2 (signal 0), transscript follows:is hyperbolic?
Note that there is an abundant supply of finitely generated non-finitely presented subgroups of 2-dimensional hyperbolic groups, given by the Rips construction MR642423.