# 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?

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Without having looked closely (I don't have the book to hand), I guess: the action of W on $\Sigma$ is cocompact if and only if W is finitely generated, which is true if and only if L is compact (which is true if and only if the cone on L is compact). – HJRW May 7 '10 at 17:11
Actually, my first statement (the action of W on $\Sigma$ is cocompact if and only if W is fg) is probably only true for freely indecomposable W. What I'm trying to say is that several of these statements probably implicitly assume that W is finitely generated. – HJRW May 7 '10 at 17:19
OK, I stand by what I said first time round. The action of W on $\Sigma$ is cocompact if and only if W is fg. (Otherwise, $\Sigma$ isn't even locally compact.) – HJRW May 7 '10 at 17:47
But doesn't $W$ act simply transitively on the vertex set of $\Sigma$ and hence collapses some vertices of $K$ into a point in $\Sigma/W$? Why is $K$ then a strict fundamental domain? – Kestutis Cesnavicius May 7 '10 at 18:30
No, I don't think W collapses any vertices of K. Here's the example I have in mind. Let L be a disjoint union of countably many points. Then W is the free product of countably many copies of Z/2, and $\Sigma$ is a locally countably-infinite tree. Each generator is a reflection in an edge of $\Sigma$. Topologically, $\Sigma/W$ is indeed the cone on L; from another point of view, it's the union of a vertex of $\Sigma$ with all the incident half-edges - in other words, a fundamental domain. Does that help? – HJRW May 7 '10 at 18:39