Quasi-isometries for certain right-angled Coxeter groups

Alexandra Edletzberger¹ 

¹Universität Wien, Austria

Groups, geometry, and dynamics, Tome 18 (2024) no. 3, pp. 1037-1098

Résumé

We construct the JSJ tree of cylinders Tc​ 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 Tc​ has two-ended edge stabilizers if and only if the defining graph does not contain a certain subdivided K4​. By use of the structure invariant of Tc​ 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.

Détail

DOI : 10.4171/ggd/779

Classification : 20F55, 20F65
Mots-clés : Coxeter groups, visual decomposition, JSJ splitting, tree of cylinders, structure invariant