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

Abstract

In this article, given two finite simplicial graphs $Γ_1$ and $Γ_2$, we state and prove a complete description of the possible morphisms $C(Γ_1) \to C(Γ_2)$ between the right-angled Coxeter groups $C(Γ_1)$ and $C(Γ_2)$. As an application, assuming that $Γ_2$ is triangle-free, we show that, if $C(Γ_1)$ is isomorphic to a subgroup of $C(Γ_2)$, then the ball of radius $8|Γ_1||Γ_2|$ in $C(Γ_2)$ contains the basis of a subgroup isomorphic to $C(Γ_1)$. 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.