Local commensurability graphs of solvable 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 $\text{GL}(2,{\mathbb{F}}_q)$.

10 pages, 2 figures

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