6 Boundaries of $CAT(0)$ spaces
Problem 6.1 (Comments) [Kim Ruane] Examples of Kleiner and Croke MR1746908, MR1924370 of non-unique boundaries are badly non-locally-connected. Is that essential in having the "flexibility" to have many boundaries? That is, does local connectedness imply uniqueness of the boundary (in the 1-ended case) for CAT(0) groups?
Background: Suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:are Gromov-hyperbolic spaces and
dvipng error! exitcode was 2 (signal 0), transscript follows:is a quasi-isometry. Then
dvipng error! exitcode was 2 (signal 0), transscript follows:extends naturally to a homeomorphism
dvipng error! exitcode was 2 (signal 0), transscript follows:. In particular, the ideal boundaries of
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:are not homeomorphic. The situation for the
dvipng error! exitcode was 2 (signal 0), transscript follows:spaces is quite different.
Definition 6.1 A group action
dvipng error! exitcode was 2 (signal 0), transscript follows:on a metric space
dvipng error! exitcode was 2 (signal 0), transscript follows:is called geometric if it is isometric, properly discontinuous and cocompact.
For a CAT(0) group
dvipng error! exitcode was 2 (signal 0), transscript follows:acting geometrically on spaces
dvipng error! exitcode was 2 (signal 0), transscript follows:, there is an induced action of
dvipng error! exitcode was 2 (signal 0), transscript follows:on the boundary
dvipng error! exitcode was 2 (signal 0), transscript follows:. For
dvipng error! exitcode was 2 (signal 0), transscript follows:-spaces
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:, the boundaries may be (a) non-homeomorphic, or (b) homeomorphic, but not
dvipng error! exitcode was 2 (signal 0), transscript follows:-equivariantly.
The Croke-Kleiner examples are torus complexes which are "combinatorially" the same but where the angle
dvipng error! exitcode was 2 (signal 0), transscript follows:between the principal circles varies. MR1746908, MR1924370 showed that these complexes
dvipng error! exitcode was 2 (signal 0), transscript follows:, which all have the same fundamental group (a right-angled Artin group, in particular), have universal covers whose boundaries are not homeomorphic when
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:. Julia Wilson showed that any two distinct values of
dvipng error! exitcode was 2 (signal 0), transscript follows:give non-homeomorphic boundaries.
Problem 6.2 (Comments) [Dani Wise] Suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:is a
dvipng error! exitcode was 2 (signal 0), transscript follows:group which does not split over a small subgroup. Does it follow that
dvipng error! exitcode was 2 (signal 0), transscript follows:is unique?
Problem 6.3 (Comments) [Dani Wise] Is the boundary well-defined for groups acting geometrically on
dvipng error! exitcode was 2 (signal 0), transscript follows:-cube complexes? More precisely, suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:are cube complexes which admit geometric actions of a group
dvipng error! exitcode was 2 (signal 0), transscript follows:. Does it follow that
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Problem 6.4 (Comments) [Ross Geoghegan] What topological invariants distinguish boundaries? In particular, what topological properties of boundaries are quasi-isometry invariants? Does something coarser than the topology stay invariant?
Remark All boundaries for a given group are shape equivalent, so cannot be distinguished by their \v{C}ech cohomology. See MR520227 for the definition of shape equivalence.
It was shown by Eric Swenson MR1802725 that for a proper cocompact
dvipng error! exitcode was 2 (signal 0), transscript follows:space
dvipng error! exitcode was 2 (signal 0), transscript follows:, the ideal boundary
dvipng error! exitcode was 2 (signal 0), transscript follows:has finite topological dimension. It was shown by Ross Geoghegan and Pedro Ontaneda MR2313068 that the topological dimension of
dvipng error! exitcode was 2 (signal 0), transscript follows:is a quasi-isometry invariant of
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Here and below a space
dvipng error! exitcode was 2 (signal 0), transscript follows:is called cocompact if
dvipng error! exitcode was 2 (signal 0), transscript follows:acts cocompactly on
dvipng error! exitcode was 2 (signal 0), transscript follows:.
A useful class of maps is called cell-like: inverse images of points are compact metrizable and each is shape equivalent to a point. (For a finite-dimensional compact subset
dvipng error! exitcode was 2 (signal 0), transscript follows:of
dvipng error! exitcode was 2 (signal 0), transscript follows:(or of any ANR) shape equivalent to a point" is equivalent to saying "
dvipng error! exitcode was 2 (signal 0), transscript follows:can be contracted to a point in any of its neighborhoods.")
Remark Cell-like maps are simple homotopy equivalences.
Problem 6.5 (Comments) [Ross Geoghegan] \label{cell-like} If
dvipng error! exitcode was 2 (signal 0), transscript follows:acts geometrically on two CAT(0) spaces, are the resulting boundaries cell-like equivalent? (That is, does there exist a space
dvipng error! exitcode was 2 (signal 0), transscript follows:with cell-like maps to each of the two spaces?)
Remark Ric Ancel, Craig Guilbault, and Julia Wilson have some examples when the answer is positive: they showed that the complexes
dvipng error! exitcode was 2 (signal 0), transscript follows:(see Croke-Kleiner examples above) are all cell-like equivalent.
Suppose that
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:are isometric cocompact properly discontinuous actions of
dvipng error! exitcode was 2 (signal 0), transscript follows:on two CAT(0) spaces.
Problem 6.6 (Comments) [Thomas Delzant] Is there a convex core for the diagonal action of
dvipng error! exitcode was 2 (signal 0), transscript follows:on
dvipng error! exitcode was 2 (signal 0), transscript follows:? (A special case is surface groups
dvipng error! exitcode was 2 (signal 0), transscript follows:with
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:corresponding to different hyperbolic structures.) If there is a convex core, can
dvipng error! exitcode was 2 (signal 0), transscript follows:(the space with cell-like maps to
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:) be taken to be the boundary of the core?
Remark [Bruce Kleiner] Convex sets are actually rare (see Remark~\ref{convexrare}), so maybe there is a different problem with better prospects.
Danny Calegari: One can try to define a new ideal boundary for
dvipng error! exitcode was 2 (signal 0), transscript follows:spaces (which is different from the visual boundary
dvipng error! exitcode was 2 (signal 0), transscript follows:) by looking at the space of all quasi-geodesics in
dvipng error! exitcode was 2 (signal 0), transscript follows:. For example, in
dvipng error! exitcode was 2 (signal 0), transscript follows:, consider all (equivalence classes of)
dvipng error! exitcode was 2 (signal 0), transscript follows:-quasi geodesics, with the compact-open topology. Varying
dvipng error! exitcode was 2 (signal 0), transscript follows:gives a filtration of the space of all quasi geodesics. Can one do interesting analysis on such a space?
Problem 6.7 (Comments) [Danny Calegari] Define a topology on the set of quasi geodesics in a (proper geodesic, or coarsely homogeneous, or cocompact)
dvipng error! exitcode was 2 (signal 0), transscript follows:space which
- has a description as an increasing union of compact metrizable spaces
has an inclusion of its visual boundary
dvipng error! exitcode was 2 (signal 0), transscript follows:
into it- is quasi-isometry invariant
- has reasonable measure classes which are quasipreserved
According to a theorem by Brian Bowditch and Gadde Swarup MR1624089,MR1412948, if
dvipng error! exitcode was 2 (signal 0), transscript follows:is a 1-ended hyperbolic group then
dvipng error! exitcode was 2 (signal 0), transscript follows:has no cut points.
For
dvipng error! exitcode was 2 (signal 0), transscript follows:a CAT(0) group, a theorem of Eric Swenson says that if
dvipng error! exitcode was 2 (signal 0), transscript follows:is a cut point, then there is an infinite-torsion subgroup of
dvipng error! exitcode was 2 (signal 0), transscript follows:fixing
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 6.8 (Comments) [Conjecture: Eric Swenson] Any
dvipng error! exitcode was 2 (signal 0), transscript follows:group has no infinite-torsion subgroups.
A Euclidean retract is a compact space that embeds into some
dvipng error! exitcode was 2 (signal 0), transscript follows:as a retract. A compact metrizable space
dvipng error! exitcode was 2 (signal 0), transscript follows:is a Z-set in
dvipng error! exitcode was 2 (signal 0), transscript follows:if it is "homotopically negligible" (for every open
dvipng error! exitcode was 2 (signal 0), transscript follows:, the inclusion
dvipng error! exitcode was 2 (signal 0), transscript follows:in
dvipng error! exitcode was 2 (signal 0), transscript follows:is a homotopy equivalence). A Z-structure on a group
dvipng error! exitcode was 2 (signal 0), transscript follows:is a pair
dvipng error! exitcode was 2 (signal 0), transscript follows:such that
dvipng error! exitcode was 2 (signal 0), transscript follows:
is a Euclidean retract,dvipng error! exitcode was 2 (signal 0), transscript follows:
is advipng error! exitcode was 2 (signal 0), transscript follows:
-set indvipng error! exitcode was 2 (signal 0), transscript follows:
,dvipng error! exitcode was 2 (signal 0), transscript follows:
admits a covering space action ofdvipng error! exitcode was 2 (signal 0), transscript follows:
withdvipng error! exitcode was 2 (signal 0), transscript follows:
compact,the set of translates of any compact set
dvipng error! exitcode was 2 (signal 0), transscript follows:
is a null sequence indvipng error! exitcode was 2 (signal 0), transscript follows:
(that is, for eachdvipng error! exitcode was 2 (signal 0), transscript follows:
there are only finitely many translates withdvipng error! exitcode was 2 (signal 0), transscript follows:
).
Finally,
dvipng error! exitcode was 2 (signal 0), transscript follows:is a boundary of
dvipng error! exitcode was 2 (signal 0), transscript follows:(or
dvipng error! exitcode was 2 (signal 0), transscript follows:-structure boundary) if there exists a
dvipng error! exitcode was 2 (signal 0), transscript follows:-structure
dvipng error! exitcode was 2 (signal 0), transscript follows:on
dvipng error! exitcode was 2 (signal 0), transscript follows:.
The above notion boundary of
dvipng error! exitcode was 2 (signal 0), transscript follows:was generalized by T.~Farrell and J.~Lafont as follows:
An EZ-boundary of a group
dvipng error! exitcode was 2 (signal 0), transscript follows:is a boundary
dvipng error! exitcode was 2 (signal 0), transscript follows:so that the action of
dvipng error! exitcode was 2 (signal 0), transscript follows:on
dvipng error! exitcode was 2 (signal 0), transscript follows:extends to topological action of
dvipng error! exitcode was 2 (signal 0), transscript follows:on
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 6.9 (Comments) [Misha Kapovich] Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a hyperbolic group and
dvipng error! exitcode was 2 (signal 0), transscript follows:be its EZ boundary. Is it true that
dvipng error! exitcode was 2 (signal 0), transscript follows:is equivariantly homeomorphic to the Gromov boundary of
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Problem 6.10 (Comments) [Mladen Bestvina] Can there be two different boundaries in the sense of
dvipng error! exitcode was 2 (signal 0), transscript follows:-structures for a group
dvipng error! exitcode was 2 (signal 0), transscript follows:that are not cell-like equivalent?
Remark Note that this problem is even open for
dvipng error! exitcode was 2 (signal 0), transscript follows:. For CAT(0) spaces, the visual boundaries are
dvipng error! exitcode was 2 (signal 0), transscript follows:-structure boundaries, so Problem~\ref{cell-like} is a special case.
Problem 6.11 (Comments) [Bruce Kleiner] Is the property of splitting over a 2-ended subgroup an invariant of Bestvina boundaries?
Some necessary conditions are known for compact, metrizable spaces
dvipng error! exitcode was 2 (signal 0), transscript follows:to be the boundary of some proper cocompact
dvipng error! exitcode was 2 (signal 0), transscript follows:space:
- should have 1,2, or infinitely many components,
dvipng error! exitcode was 2 (signal 0), transscript follows:
is finite dimensional (Theorem of Swenson),dvipng error! exitcode was 2 (signal 0), transscript follows:
has nontrivial top \v{C}ech cohomology (Geoghegan--Ontaneda MR2313068).
In the case when
dvipng error! exitcode was 2 (signal 0), transscript follows:admits a cocompact free(?) action by a discrete subgroup of isometries, one necessary condition is due to Bestvina: the dimension of every nonempty open set
dvipng error! exitcode was 2 (signal 0), transscript follows:is equal to the dimension of
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 6.12 (Comments) [Ross Geoghegan]\label{rossq} Extend these lists, or give a complete classification.
Problem 6.13 (Comments) [Kevin Whyte]\label{why} Does every CAT(0) group have finite asymptotic dimension?