Examples, Snippets

A virtually free RACG

(Thanks to Harry Richman for bringing this example up.)

The free group \langle a, b \rangle and the right-angled Coxeter group \rangle a, b, c, d | a^2, b^2, c^2, d^2 \rangle have the same Cayley graph—the infinite 4-valent tree—; hence they are quasi-isometric. We note here that quasi-isometries are not required to group homomorphisms, and indeed the one we have here—an “identity map” of sorts (in an admittedly very loose and imprecise sense) between Cayley graphs—is not.

By results of Stallings and Dunwoody involving ends of groups ad accessibility (although Chapter 20 of Drutu-Kapovich’s draft manuscript seems to be the only reference I can find for this online), free groups are quasi-isometrically rigid, and hence, since finite-index subgroups of free groups are free, the RACG is virtually free (!)

With a little more thought we can explicitly exhibit a finite-index free subgroup: the subgroup of the RACG generated by ab, ac and ad is a nonabelian free group on three generators; we can verify it has finite index via a covering space argument.

Now I wonder if there is a finite-index F_2-subgroup of the RACG as well, so that the commensurability need not involve passing to finite-index subgroups on both sides. It seems like there shouldn’t be, although I can’t quite prove this yet.



CAT(0) not a quasi-isometry invariant of groups

It is easy enough to produce examples of pairs of spaces which are quasi-isometric, but only one of which is CAT(0): the CAT(k) inequality applies at every scale, but quasi-isometries do not allow any control over what happens at small scales. For instance, we could quasi-isometrically map Euclidean space into a space with regions of (arbitrarily large) positive curvature concentrated inside bounded balls, in which small triangles will fail to satisfy the CAT(0) condition.

It is a little trickier to produce examples of pairs of groups which are quasi-isometric, but only one of which is CAT(0), since now the CAT(0) condition applies not directly to the group, but refers to (the existence of) a CAT(0) space on which the group acts geometrically and cocompactly (and hence a fortiori by semisimple isometries), and we have, at first blush, great freedom to choose what that space might be, depending on what the group is.

Here are two constructions that work, both of which are pointed out in Remark 1 of this paper of Piggott-Ruane-Walsh (although one of them—the second example below—somewhat badly).

Fundamental groups of graph manifolds

One example follows from the work of Kapovich-Leeb on fundamental groups of graph manifolds. Kapovich-Leeb proved (Theorem 1.1) that whenever M is a Haken (i.e. contains an essential properly embedded surface) Riemmanian 3-manifold with \chi(M) = 0, \pi(M) is quasi-isometric to a fundamental group of a compact non-positively curved 3-manifold; groups of the latter description are always CAT(0).

On the other hand, Leeb showed (Example 4.2) that there exist Haken 3-manifolds with \chi(M) = 0—more specifically, closed graph manifolds with no atoroidal components  (i.e. all of whose geometric components are Seifert-fibred) and empty boundary—which do not admit metrics of nonpositive curvature; decomposing the manifold we see that there exist spherical regions in M, and then it follows that the fundamental groups \pi_1(M) of such manifolds M cannot be CAT(0).

Mapping class groups (and unit tangent bundles)

A different example can be found by messing around with mapping class groups; the following is taken from p. 258 of Bridson and Haefliger’s Metric Spaces of Nonpositive Curvature (the top half of the page, not the bottom half.)

Let K be the central extension 1 \to \langle T_c \rangle \to K \to \pi_1(\Sigma_2) \to 1 defined as the kernel of the natural homomorphisms \mathrm{Mod}(\Sigma_2^b) \to \mathrm{Mod}(\Sigma_2^p) \to \mathrm{Mod}(\Sigma_2), where \Sigma_2 is the closed genus-2 surface, \Sigma_2^b denotes the genus-2 surface with a single boundary component c \cong S^1 (and T_c denotes the Dehn twist about c), and \Sigma_2^p denotes the genus-2 surface with a single puncture.

By an unpublished observation of Geoffrey Mess (looking at the geometry of the mapping classes), K is the fundamental group of the unit tangent bundle S\Sigma_2 of \Sigma_2, and hence a cocompact lattice in \widetilde{\mathrm{PSL}_2\mathbb{R}}, but this last does not contain the fundamental group of any closed hyperbolic surface. But, on the other hand, any finite-index subgroup of \pi_1(\Sigma_2) is, by covering space theory, the fundamental group of a closed hyperbolic surface; and now we may conclude that our central extension cannot be split, even after passing to a finite-index subgroup of \pi_1(\Sigma_2).

Now we have the following general result on isometries of CAT(0) spaces: any group of isometries of a CAT(0) space which contains a central A \cong \mathbb{Z}^n subgroup contains a finite-index subgroup which has A as a direct factor.

Hence we conclude that our K cannot be (isomorphic to) a group of semisimple isometries of any CAT(0) space, i.e. K is not a CAT(0) group.

On the other hand, Epstein, Gersten (and Mess? So claims Misha Kapovich) independently proved that \pi_1(S\Sigma_2) is quasi-isometric to \pi_1(\Sigma_2) \times \mathbb{Z}, which is certainly CAT(0), e.g. being the product of two CAT(0) groups. This—that two of the Thurston model geometries, \widetilde{\mathrm{PSL}_2\mathbb{R}} and \mathbb{H}^2 \times \mathbb{R}, are quasi-isometric to each other—is apparently also a [multiply] unpublished observation—trying to track down a reference was annoyingly tricky. Gromov’s Metric Structures for Riemannian and Non-Riemannian Spaces contains an account.