Quasi-Isometries for certain Right-Angled Coxeter Groups

DOI10.4171/GGD/779arXiv2112.10463MaRDI QIDQ6505295

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}

. 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.

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