On Commensurizer Growth

TL;DR

This paper introduces and analyzes the concept of commensurizer growth, an asymptotic invariant for pairs of groups and subgroups, with computations mainly focused on locally compact groups and lattices.

Contribution

It defines the new invariant of commensurizer growth and provides explicit calculations for various examples, especially in the context of topological groups and lattices.

Findings

Computed commensurizer growth for several group pairs

Established properties and behaviors of the invariant in different settings

Provided insights into the structure of lattices in topological groups

Abstract

We study new asymptotic invariant of a pair consisting of a group and a subgroup, which we call Commensurizer Growth. We compute the commensurizer growth for several examples, concentrating mainly on the case of a locally compact topological group and a lattice inside it.