A question on a Davis complex of a Coxeter group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T07:31:03Z http://mathoverflow.net/feeds/question/23877 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23877/a-question-on-a-davis-complex-of-a-coxeter-group A question on a Davis complex of a Coxeter group Kestutis Cesnavicius 2010-05-07T16:57:11Z 2010-05-07T16:57:11Z <p>Let us have a look at p. 64 of M. Davis book "The Geometry and Topology of Coxeter Groups". The discussion preceeding Definition 5.1.3. shows that $\mathcal{U}(G, X)/G$ is homeomorphic to $X$. Theorem 7.2.4. says that $\mathcal{U}(W, K)$ is $W$-equivariantly homeomorphic to the Davis complex $\Sigma$. So, $\Sigma/W$ is homeomorphic to $K$. $K$ is the cone on the barycentric subdivision of the nerve $L$. $L$ can have topological type of any polyhedron. So $K$ can be a cone on any polyhedron (up to homeomorphism). But the action of $W$ on $\Sigma$ is cocompact (p. 4, bottom). So $\Sigma/W$ is compact, i.e., $K$ is compact. So a cone on any polyhedron is compact. What's wrong?</p>