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 ofW 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 the results of A. Kaul and N. Brady, J.P. McCammond, B. Mühlherr and W.D. Neumann.