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 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.

arXiv comment50 pages

References in corpus (8)

• JSJ decompositions of groups
• Trees of cylinders and canonical splittings
• Quasi-isometries Between Groups with Two-Ended Splittings
• Global Structural Properties of Random Graphs
• Quasi-isometry classification of RAAGs that split over cyclic subgroups
• On the coarse geometry of certain right-angled Coxeter groups
• Computing JSJ decompositions of hyperbolic groups
• Commensurators of parabolic subgroups of Coxeter groups

Cited by in corpus (3)

• Visual right-angled Artin subgroups of two-dimensional right-angled Coxeter groups
• RAAGedy right-angled Coxeter groups II: in the quasiisometry class of the tree RAAGs
• Finite-State Machines for Horospheres in Hyperbolic Right-Angled Coxeter Groups