2112.10463Dec 20, 202150 pages

Quasi-Isometries for certain Right-Angled Coxeter Groups

Alexandra Edletzberger

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 given for 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 $K_4$

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. By use of the structure invariant of

T_c

introduced by Cashen and Martin, 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.