# Is $SL(n,\mathbb{Z})$ a CAT(0) group?

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.

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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

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