Stabilizers in Coxeter groups with a transpositional action

We study simply-laced Coxeter groups whose Coxeter diagram is the line graph $\mathcal{L}\mathrm{\Gamma }$ of some simple graph $\mathrm{\Gamma }$; such groups act on the set of vertices of the graph by transpositions. Our goal is to describe the point stabilizer of this action in detail. For any tree $\mathrm{\Sigma }$, we show that the stabilizer is a finite-index reflection subgroup, and identify its Coxeter diagram, thereby discovering finite-index embeddings of Coxeter groups $W({\mathcal{L}}^{[2]}\mathrm{\Sigma })\hookrightarrow W(\mathcal{L}\mathrm{\Sigma })$ for any tree $\mathrm{\Sigma }$, including infinitely many cases where the stabilizer itself is simply-laced.