MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is it possible to find a CAT(0) space on which the matrix group $SL(n,\mathbb{Z})$ acts properly discontinuously and cocompactly? Note: when the cocompactness is dropped , it is possible.

share|cite|improve this question
the filling of $n-2$-spheres in $SL_n(\mathbf{Z})$ for $n\ge 3$ has exponential area. So $SL_n(\mathbf{Z})$ is not quasi-isometric to a CAT(0) space (and, in the same vein, its asymptotic cones have nontrivial $\pi_{n-2}$). Corollary: it has no proper cocompact action on a CAT(0) space. – YCor Dec 1 '13 at 12:11

If n=2, yes: it acts on its Bass-Serre tree.

If n>2, no: your group contains distorted elements, i.e. elements conjugated to a proper power of themselves (look at unipotent matrices).

Such an element will have zero displacement length, which is impossible for an infinite order element in a group acting discretely cocompactly.

For this and much more, see the monograph of Bridson and Haefliger.

For even more restrictions on SL_n actions, see Theorem 1.14 in:

Caprace-Monod, Isometry groups of non-positively curved spaces: structure theory Journal of Topology 2 No. 4 (2009), 661–700

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.