simplicial stuff - A question on a Davis complex of a Coxeter group

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?