A new lattice invariant for lattices in totally disconnected locally compact groups

TL;DR

This paper introduces the bounded conjugacy rank, a new lattice invariant for totally disconnected locally compact groups, and demonstrates its invariance across all lattices within such groups, supported by examples.

Contribution

The paper defines a novel lattice invariant called the bounded conjugacy rank and proves its invariance for lattices in sigma compact totally disconnected locally compact groups.

Findings

Bounded conjugacy rank is a lattice invariant.

The invariant applies to sigma compact totally disconnected locally compact groups.

Examples illustrate the concept and its applications.

Abstract

We introduce and explore a natural rank for totally disconnected locally compact groups called the bounded conjugacy rank. This rank is shown to be a lattice invariant for lattices in sigma compact totally disconnected locally compact groups; that is to say, for a given sigma compact totally disconnected locally compact group, some lattice has bounded conjugacy rank n if and only if every lattice has bounded conjugacy rank n. Several examples are then presented.