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

Abstract

In this article, given two finite simplicial graphs ${\Gamma }_1$ and ${\Gamma }_2$, we state and prove a complete description of the possible morphisms $C({\Gamma }_1)\to C({\Gamma }_2)$ between the right-angled Coxeter groups $C({\Gamma }_1)$ and $C({\Gamma }_2)$. As an application, assuming that ${\Gamma }_2$ is triangle-free, we show that, if $C({\Gamma }_1)$ is isomorphic to a subgroup of $C({\Gamma }_2)$, then the ball of radius $8|{\Gamma }_1||{\Gamma }_2|$ in $C({\Gamma }_2)$ contains the basis of a subgroup isomorphic to $C({\Gamma }_1)$. 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.

arXiv comment40 pages, 6 figures. Theorems 1.3 and 1.6 added, and minor modifications