New York Journal of Mathematics
Volume 24 (2018), 429-442


Khalid Bou-Rabee and Daniel Studenmund

The topology of local commensurability graphs

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Published: July 26, 2018
Keywords: commensurability, nilpotent groups, free groups, very large groups.
Subject: Primary: 20E26 and 20E15; Secondary: 20B99 and 20F18.

We initiate the study of the p-local commensurability graph of a group, where p is a prime.This graph has vertices consisting of all finite-index subgroups of a group, where an edge is drawn between A and B if [A: A ∩ B] and [B: A ∩ B] are both powers of p. We show that any component of the p-local commensurability graph of a group with all nilpotent finite quotients is complete. Further, this topological criterion characterizes such groups. In contrast to this result, we show that for any prime p the p-local commensurability graph of any large group (e.g. a nonabelian free group or a surface group of genus two or more or, more generally, any virtually special group) has geodesics of arbitrarily long length.


D.S. supported in part by NSF grants DMS-1246989 and DMS-1547292

Author information

Khalid Bou-Rabee
Department of Mathematics, CCNY CUNY

Daniel Studenmund
Department of Mathematics, University of Notre Dame