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 \(C(\Gamma _1) \rightarrow C(\Gamma _2)\) 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.