TITLE: Is $SL(n,\mathbb{Z})$ a CAT(0) group?
QUESTION [11 upvotes]: 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.

REPLY [24 votes]: 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