User anon - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T14:39:39Z http://mathoverflow.net/feeds/user/13877 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82058/is-sln-mathbbz-a-cat0-group/82065#82065 Answer by Anon for Is \$SL(n,\mathbb{Z})\$ a CAT(0) group? Anon 2011-11-28T09:35:06Z 2011-11-28T09:35:06Z <p>If n=2, yes: it acts on its Bass-Serre tree.</p> <p>If n>2, no: your group contains distorted elements, i.e. elements conjugated to a proper power of themselves (look at unipotent matrices).</p> <p>Such an element will have zero displacement length, which is impossible for an infinite order element in a group acting discretely cocompactly.</p> <p>For this and much more, see the monograph of Bridson and Haefliger.</p> <p>For even more restrictions on SL_n actions, see Theorem 1.14 in:</p> <p>Caprace-Monod, Isometry groups of non-positively curved spaces: structure theory Journal of Topology 2 No. 4 (2009), 661–700</p> http://mathoverflow.net/questions/78437/bounded-cohomology-of-subgroups-of-groups/78442#78442 Answer by Anon for bounded cohomology of subgroups of groups Anon 2011-10-18T10:52:46Z 2011-10-18T10:52:46Z <p>In general, no. There is not even a natural map (in general) in the direction you want.</p> <p>There is a natural map in the opposite direction, namely restriction, and this is sometimes an embedding, but not always. See chapter 8.6 in "Continuous bounded cohomology of locally compact groups", Lecture Notes in Mathematics 1758</p> http://mathoverflow.net/questions/59428/hyperbolic-isometries-in-cocompact-hadamard-i-e-cat0-proper-simply-connected/59437#59437 Answer by Anon for Hyperbolic isometries in cocompact Hadamard (i.e. cat(0) proper simply connected) spaces Anon 2011-03-24T15:00:52Z 2011-03-24T15:00:52Z <p>There is an answer, under perhaps some conditions, in Section 6.C of</p> <p>Caprace, Pierre-Emmanuel; Monod, Nicolas Isometry groups of non-positively curved spaces: structure theory. J. Topol. 2 (2009), no. 4, 661-700.</p> <p>Regarding the notations in that reference: notice that if X is any proper CAT(0) space, then the group G=Isom(X) will automaticall act properly on X.</p> http://mathoverflow.net/questions/83127/examples-of-cat0-groups/83129#83129 Comment by Anon Anon 2011-12-15T10:01:46Z 2011-12-15T10:01:46Z To be fair, it is not a trivial fact that a simply connected Riemannian manifold whose sectional curvatures are all nonpositive is CAT(0). http://mathoverflow.net/questions/59428/hyperbolic-isometries-in-cocompact-hadamard-i-e-cat0-proper-simply-connected/59437#59437 Comment by Anon Anon 2011-04-05T14:02:52Z 2011-04-05T14:02:52Z This is because they give the definition in the particular case of discrete groups. The &quot;true&quot; definition&quot; replaces &quot;finite&quot; by &quot;compact&quot;. But in general, the topology of the group of isometries is defined in such a way that the action is proper almost by definition. So this is definitely not a problem.