Local commensurability graphs of solvable groups

TL;DR

This paper studies the structure of p-local commensurability graphs in solvable groups, showing bounded diameters for metabelian groups and classifying components for certain matrix groups.

Contribution

It establishes diameter bounds for components of p-local commensurability graphs in metabelian groups and provides a complete classification for upper triangular matrix groups.

Findings

Bounded diameter of 4 for metabelian groups' graph components.

No universal diameter bound for all finite solvable groups.

Complete classification of components for upper triangular matrix groups.

Abstract

The commensurability index between two subgroups $A, B$ of a group $G$ is $[A : A \cap B] [B : A\cap B]$. This gives a notion of distance amongst finite-index subgroups of $G$, which is encoded in the p-local commensurability graphs of $G$. We show that for any metabelian group, any component of the $p$-local commensurabilty graph of $G$ has diameter bounded above by 4. However, no universal upper bound on diameters of components exists for the class of finite solvable groups. In the appendix we give a complete classification of components for upper triangular matrix groups in $\text{GL}(2, \mathbb{F}_q)$.