Quasi-isometries for certain right-angled Coxeter groups

Abstract

We construct the JSJ tree of cylinders $T_c$ for finitely presented, one-ended, two-dimensional right-angled Coxeter groups (RACGs) splitting over two-ended subgroups in terms of the defining graph of the group, generalizing the visual construction by Dani and Thomas [J. Topol. 10 (2017), 1066–1106] given for certain hyperbolic RACGs. Additionally, we prove that $T_c$ has two-ended edge stabilizers if and only if the defining graph does not contain a certain subdivided $K_4$. By use of the structure invariant of $T_c$ introduced by Cashen and Martin [Math. Proc. Cambridge Philos. Soc. 162 (2017), 249–291], we obtain a quasi-isometry invariant of these RACGs, essentially determined by the defining graph. Furthermore, we refine the structure invariant to make it a complete quasi-isometry invariant in case the JSJ decomposition of the RACG does not have any rigid vertices.