Geodesic
Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups
Pallavi Dani1 ; Emily Stark2 ; Anne Thomas3 orcid
1Louisiana State University, Baton Rouge, USA
2Technion - Israel Institute of Technology, Haifa, Israel
3University of Sydney, Australia
Groups, geometry, and dynamics, Tome 12 (2018) no. 4, pp. 1273-1341

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DOI

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 Θ-graphs and cycles of generalized Θ-graphs, and geometric amalgams of free groups whose JSJ graphs are trees of diameter ≤4. 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.
DOI : 10.4171/ggd/469
Classification : 57-XX, 20-XX
Mots-clés : Commensurability, right-angled Coxeter groups, amalgams

Pallavi Dani  1   ; Emily Stark  2   ; Anne Thomas  3

1 Louisiana State University, Baton Rouge, USA
2 Technion - Israel Institute of Technology, Haifa, Israel
3 University of Sydney, Australia 
Pallavi Dani; Emily Stark; Anne Thomas. Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups. Groups, geometry, and dynamics, Tome 12 (2018) no. 4, pp. 1273-1341. doi: 10.4171/ggd/469
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     title = {Commensurability for certain right-angled {Coxeter} groups and geometric amalgams of free groups},
     journal = {Groups, geometry, and dynamics},
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     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/469/}
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