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 $X, Y$ are Gromov-hyperbolic spaces and $f: X\to Y$ is a quasi-isometry. Then $f$ extends naturally to a homeomorphism $\geo f: \geo X\to \geo Y$. In particular, the ideal boundaries of $X$ and $Y$ are not homeomorphic. The situation for the $CAT(0)$ spaces is quite different.

Definition 6.1 A group action $G\acts X$ on a metric space $X$ is called geometric if it is isometric, properly discontinuous and cocompact.

For a CAT(0) group $G$ acting geometrically on spaces $X_i$, there is an induced action of $G$ on the boundary $\geo X_i$. For $G$-spaces $X_1$ and $X_2$, the boundaries may be (a) non-homeomorphic, or (b) homeomorphic, but not $G$-equivariantly.

The Croke-Kleiner examples are torus complexes which are "combinatorially" the same but where the angle $\alpha$ between the principal circles varies. MR1746908, MR1924370 showed that these complexes $K_\alpha$, which all have the same fundamental group (a right-angled Artin group, in particular), have universal covers whose boundaries are not homeomorphic when $\alpha=\pi/2$ and $\alpha\neq \pi/2$. Julia Wilson showed that any two distinct values of $\alpha$ give non-homeomorphic boundaries.

Problem 6.2 (Comments) [Dani Wise] Suppose that $G$ is a $CAT(0)$ group which does not split over a small subgroup. Does it follow that $\geo G$ is unique?

Problem 6.3 (Comments) [Dani Wise] Is the boundary well-defined for groups acting geometrically on $CAT(0)$-cube complexes? More precisely, suppose that $X_1, X_2$ are cube complexes which admit geometric actions of a group $G$. Does it follow that $\geo X_1=\geo X_2$?

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 $CAT(0)$ space $X$, the ideal boundary $\geo X$ has finite topological dimension. It was shown by Ross Geoghegan and Pedro Ontaneda MR2313068 that the topological dimension of $\geo X$ is a quasi-isometry invariant of $X$.

Here and below a space $X$ is called cocompact if $Isom(X)$ acts cocompactly on $X$.

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 $Y$ of $\R^n$ (or of any ANR) shape equivalent to a point" is equivalent to saying "$Y$ 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 $G$ acts geometrically on two CAT(0) spaces, are the resulting boundaries cell-like equivalent? (That is, does there exist a space $Z$ 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 $K_\alpha$ (see Croke-Kleiner examples above) are all cell-like equivalent.

Suppose that $G\acts X_i$, $i=1, 2$ are isometric cocompact properly discontinuous actions of $G$ on two CAT(0) spaces.

Problem 6.6 (Comments) [Thomas Delzant] Is there a convex core for the diagonal action of $G$ on $X_1\times X_2$? (A special case is surface groups $G$ with $X_1$ and $X_2$ corresponding to different hyperbolic structures.) If there is a convex core, can $Z$ (the space with cell-like maps to $X_1$ and $X_2$) 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 $CAT(0)$ spaces (which is different from the visual boundary $\geo X$) by looking at the space of all quasi-geodesics in $X$. For example, in $\R^2$, consider all (equivalence classes of) $K$-quasi geodesics, with the compact-open topology. Varying $K$ 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) $CAT(0)$ space which

  1. has a description as an increasing union of compact metrizable spaces

  2. has an inclusion of its visual boundary $\geo X$ into it

  3. is quasi-isometry invariant
  4. has reasonable measure classes which are quasipreserved

According to a theorem by Brian Bowditch and Gadde Swarup MR1624089,MR1412948, if $G$ is a 1-ended hyperbolic group then $\geo G$ has no cut points.

For $G$ a CAT(0) group, a theorem of Eric Swenson says that if $c\in\geo G$ is a cut point, then there is an infinite-torsion subgroup of $G$ fixing $c$.

Problem 6.8 (Comments) [Conjecture: Eric Swenson] Any $CAT(0)$ group has no infinite-torsion subgroups.

A Euclidean retract is a compact space that embeds into some $\R^n$ as a retract. A compact metrizable space $Z$ is a Z-set in $\tilde{X}$ if it is "homotopically negligible" (for every open $U\subset \tilde{X}$, the inclusion $U\setminus Z$ in $U$ is a homotopy equivalence). A Z-structure on a group $G$ is a pair $(\tilde{X},Z)$ such that

  1. $\tilde{X}$ is a Euclidean retract,

  2. $Z$ is a $Z$-set in $\tilde{X}$,

  3. $X:=\tilde{X}\setminus Z$ admits a covering space action of $G$ with $X/G$ compact,

  4. the set of translates of any compact set $K\subset X$ is a null sequence in $\tilde{X}$ (that is, for each $\epsilon>0$ there are only finitely many translates with ${\rm diam}>\epsilon$).

Finally, $Z$ is a boundary of $G$ (or $Z$-structure boundary) if there exists a $Z$-structure $(\tilde{X},Z)$ on $G$.

The above notion boundary of $G$ was generalized by T.~Farrell and J.~Lafont as follows:

An EZ-boundary of a group $G$ is a boundary $Z=\partial_{EZ}G$ so that the action of $G$ on $X$ extends to topological action of $G$ on $Z$.

Problem 6.9 (Comments) [Misha Kapovich] Let $G$ be a hyperbolic group and $\D_{EZ} G$ be its EZ boundary. Is it true that $\D_{EZ} G$ is equivariantly homeomorphic to the Gromov boundary of $G$?

Problem 6.10 (Comments) [Mladen Bestvina] Can there be two different boundaries in the sense of $Z$-structures for a group $G$ that are not cell-like equivalent?

Remark Note that this problem is even open for $\Z^n$. For CAT(0) spaces, the visual boundaries are $Z$-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 $X$ to be the boundary of some proper cocompact $CAT(0)$ space:

  1. should have 1,2, or infinitely many components,

  2. $X$ is finite dimensional (Theorem of Swenson),

  3. $X$ has nontrivial top \v{C}ech cohomology (Geoghegan--Ontaneda MR2313068).

In the case when $X$ 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 $U \subset X$ is equal to the dimension of $X$.

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?

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Boundaries/Section6 (last edited 2008-04-27 21:40:35 by RickScott)