The topology of local commensurability graphs

Abstract

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 $[A:A\cap B]$ and $[B:A\cap B]$ 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.

12 pages

References in corpus (3)

• The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces. With an appendix by Khalid Bou-Rabee
• The Prime Index Graph of a Group
• Divisibility and Laws in Finite Simple Groups

Cited by in corpus (1)

• Local commensurability graphs of solvable groups