The topology of local commensurability graphs

TL;DR

This paper introduces the concept of p-local commensurability graphs for groups, characterizes groups with complete components based on nilpotent quotients, and explores the complex structure of these graphs in large groups.

Contribution

It defines p-local commensurability graphs, characterizes groups with complete components via nilpotent quotients, and analyzes the graph structure in large groups.

Findings

Groups with all nilpotent finite quotients have complete p-local commensurability graph components.

The p-local commensurability graph of large groups contains arbitrarily long geodesics.

The topological criterion characterizes groups with complete components in the graph.

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.