 

Khalid BouRabee and
Daniel Studenmund
The topology of local commensurability graphs view print


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


Abstract
We initiate the study of the plocal commensurability graph of a group, where p is a prime.This graph has vertices consisting of all finiteindex 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 plocal 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 plocal 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.


Acknowledgements
D.S. supported in part by NSF grants DMS1246989 and DMS1547292


Author information
Khalid BouRabee Department of Mathematics, CCNY CUNY
khalid.math@gmail.com
Daniel Studenmund Department of Mathematics, University of Notre Dame
dstudenm@nd.edu

