Commensurability growths of algebraic groups

TL;DR

This paper investigates the growth of subgroups commensurable with a fixed lattice in algebraic groups, providing precise estimates and relating it to subgroup growth and an invariant depending only on the group.

Contribution

It introduces exact estimates for commensurability growth in Chevalley groups and links it to subgroup growth and a computable group invariant.

Findings

Derived precise bounds for commensurability growth functions.

Connected commensurability growth to subgroup growth and a specific invariant.

Enhanced understanding of subgroup structures in algebraic groups.

Abstract

Fixing a subgroup $\Gamma$ in a group $G$, the full commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ of $G$ with $[\Gamma: \Gamma \cap \Delta][\Delta : \Gamma \cap \Delta] \leq n$. For pairs $\Gamma \leq G$, where $G$ is a Chevalley group scheme defined over $\mathbb{Z}$ and $\Gamma$ is an arithmetic lattice in $G$, we give precise estimates for the full commensurability growth, relating it to subgroup growth and a computable invariant that depends only on $G$.