Coxeter systems with two-dimensional Davis-Vinberg complexes

In this paper, we study Coxeter systems with two-dimensional Davis-Vinberg complexes. We show that for a Coxeter group $W$, if $(W,S)$ and $(W,S')$ are Coxeter systems with two-dimensional Davis-Vinberg complexes, then there exists $S''\subset W$ such that $(W,S'')$ is a Coxeter system which is isomorphic to $(W,S)$ and the sets of reflections in $(W,S'')$ and $(W,S')$ coincide. Hence the Coxeter diagrams of $(W,S)$ and $(W,S')$ have the same number of vertices, the same number of edges and the same multiset of edge-labels. This is an extension of results of A.Kaul and N.Brady, J.P.McCammond, B.M\"uhlherr and W.D.Neumann.