Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups

  • Pallavi Dani

    Louisiana State University, Baton Rouge, USA
  • Emily Stark

    Technion - Israel Institute of Technology, Haifa, Israel
  • Anne Thomas

    University of Sydney, Australia

Abstract

We give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized -graphs and cycles of generalized -graphs, and geometric amalgams of free groups whose JSJ graphs are trees of diameter . We also show that if a geometric amalgamof free groups has JSJ graph a tree, then it is commensurable to a right-angled Coxeter group, and give an example of a geometric amalgam of free groups which is not quasi-isometric (hence not commensurable) to any group which is finitely generated by torsion elements. Our proofs involve a new geometric realization of the right-angled Coxeter groups we consider, such that covers corresponding to torsion-free, finite-index subgroups are surface amalgams.

Cite this article

Pallavi Dani, Emily Stark, Anne Thomas, Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups. Groups Geom. Dyn. 12 (2018), no. 4, pp. 1273–1341

DOI 10.4171/GGD/469