infinite dimensional CAT(0) groups - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:45:22Z http://mathoverflow.net/feeds/question/84957 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84957/infinite-dimensional-cat0-groups infinite dimensional CAT(0) groups HenrikRüping 2012-01-05T13:23:59Z 2012-03-26T20:46:17Z <p>Usually a CAT(0) group is defined to be a group acting properly isometrically and cocompactly on a CAT(0) space, but I would like to consider only those groups that act properly, isometrically and cocompactly on a finite-dimensional CAT(0) space.</p> <p>So is there a group I have to leave out?</p> <p>Not every CAT(0) space with a proper isometric cocompact group action is finite-dimensional. For example the trivial group acts on the compact CAT(0)-space \$[0;1]^\mathbb{N}\$. </p> http://mathoverflow.net/questions/84957/infinite-dimensional-cat0-groups/84966#84966 Answer by Alain Valette for infinite dimensional CAT(0) groups Alain Valette 2012-01-05T15:16:22Z 2012-01-05T16:53:44Z <p>I think you will miss the notorious Thompson's group \$F\$: it acts properly isometrically on a \$CAT(0)\$ cube complex (see Dan Farley, <a href="http://www.users.muohio.edu/farleyds/Far1.pdf" rel="nofollow">http://www.users.muohio.edu/farleyds/Far1.pdf</a>); but it cannot act properly isometrically on a finite-dimensional complex, as this would make it of finite cohomological dimension.</p> <p>EDIT: Indeed as Mark pointed out, the action of \$F\$ on this \$CAT(0)\$-cube complex is NOT co-compact, hence my answer should be discarded. Don't trust MO too much. </p> http://mathoverflow.net/questions/84957/infinite-dimensional-cat0-groups/92308#92308 Answer by Misha for infinite dimensional CAT(0) groups Misha 2012-03-26T20:46:17Z 2012-03-26T20:46:17Z <p>First, I suppose that by <em>proper action</em> you mean the one in the sense of Bridson and Haefliger, otherwise you would have to regard \${\mathbb R}\$ as a \$CAT(0)\$ groups. Now, it follows from Eric Swenson's paper "A cut point theorem for CAT(0) groups" (Journal of Diff. Geometry, 1999) that the ideal boundary of the \$CAT(0)\$ space \$X\$ (on which a group \$G\$ acts geometrically) is finite-dimensional. This suffices for many practical purposes. For instance, it follows (from Bestvina's work) that \$G\$ has finite cohomological dimension over \${\mathbb Q}\$ and, if you consider torsion-free groups, over \${\mathbb Z}\$ as well. (This immediately excludes Thompson's group, etc.) In particular, geometric dimension of \$G\$ is finite, \$G\$ has finite type, etc. From this you can make pretty much the same algebraic conclusions about \$G\$ as in the case when \$G\$ acts geometrically on a finite-dimensional \$CAT(0)\$ space. Thus, in the torsion-free case, I do not think you are missing (or gaining) much by restricting to finite-dimensional \$CAT(0)\$ spaces. (For instance, I do not see how assuming finite dimension of \$X\$ would help with proving that \$G\$ has finite asymptotic dimension.) </p> <p>I am not sure what happens in the case of groups with torsion: It is conjectured by Swenson that a \$CAT(0)\$ group \$G\$ cannot contain infinite torsion subgroups. Maybe it would be easier to exclude some infinite torsion subgroups (say, the infinite permutation group) using the assumption that \$G\$ acts geometrically on a finite-dimensional \$CAT(0)\$ space, but I do not see how. </p> <p>Swenson's work had a follow-up paper by Geoghegan and Ontaneda <a href="http://arxiv.org/abs/math/0407506" rel="nofollow">http://arxiv.org/abs/math/0407506</a> where they weaken some of his assumptions and strengthen some of his conclusions. </p> <p>Note: In view of Swenson's result it is tempting to say: Take the closed convex hull (in \$X\$) of the ideal boundary of the \$CAT(0)\$ space \$X\$ and show that it is finite-dimensional. It might work, but, in general, convex hulls in \$CAT(0)\$ spaces tend to be much bigger than expected. </p>