Are all these groups CAT(0) groups? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:13:35Z http://mathoverflow.net/feeds/question/37968 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37968/are-all-these-groups-cat0-groups Are all these groups CAT(0) groups? HenrikRüping 2010-09-07T11:58:20Z 2010-09-08T06:04:31Z <p>Given a geodesic metric space $X$ together with a choice of midpoints $m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$). Assume furthermore, that the following nonpositive curvature condition is satisfied:</p> <p>$d(m(x,y),m(x,z))\le \frac{d(y,z)}{2}$ for all $x,y,z\in X$ . This is just a special case of the CAT(0) inequality for the "triangle" $x,y,z$. Lets call such a space a M-space.</p> <p>Such a space needn't be <a href="http://en.wikipedia.org/wiki/CAT%2528k%2529_space" rel="nofollow">CAT(0)</a>, as the example $(\mathbb{R}^n,d^1)$ shows, where $d^1$ is the $l^1$ metric. The choice of midpoints is given by $m(x,y)=\frac{x+y}{2}$. It also needn't be unique geodesic.</p> <p>But this space can be equipped with another metric, that makes it a CAT(0) space.</p> <p>So my question is: Is every group, that acts properly, isometrically and cocompactly on a M-space already a CAT(0)-group?</p>