Morphisms between right-angled Coxeter groups and the embedding problem in dimension two

Abstract

In this article, given two finite simplicial graphs Γ1 and Γ2, we state and prove a complete description of the possible morphisms C(Γ1) → C(Γ2) between the right-angled Coxeter groups C(Γ1) and C(Γ2). As an application, assuming that Γ2 is triangle-free, we show that, if C(Γ1) is isomorphic to a subgroup of C(Γ2), then the ball of radius 8|Γ1||Γ2| in C(Γ2) contains the basis of a subgroup isomorphic to C(Γ1). This provides an algorithm determining whether or not, among two given two-dimensional right-angled Coxeter groups, one is isomorphic to a subgroup of the other.