Geometric invariants of TDLC completions

TL;DR

This paper explores the geometric invariants of totally disconnected locally compact (TDLC) completions of Hecke pairs, revealing how their $\Sigma$-sets relate to those of the original group under certain conditions, with applications to specific group classes.

Contribution

It generalizes the understanding of how $\Sigma$-sets behave under TDLC completions of Hecke pairs, extending previous results on Schlichting completions.

Findings

$\Sigma$-sets of TDLC completions relate to original groups' $\Sigma$-sets under compactness conditions.

Application to Baumslag-Solitar groups and upper triangular matrix groups.

Provides a framework for analyzing geometric invariants in TDLC group completions.

Abstract

Recently, Bonn and Sauer showed that, from the point of view of compactness properties, the Schlichting completion of a Hecke pair $(\Gamma,\Lambda)$ behaves precisely as if it were the quotient of $\Gamma$ by $\Lambda$. Motivated by this result, we prove that a similar phenomenon holds for the $\Sigma$-sets. More generally, we relate the $\Sigma$-sets of every TDLC completion of a Hecke pair $(\Gamma,\Lambda)$ to the $\Sigma$-sets of $\Gamma$ whenever $\Lambda$ satisfies suitable compactness properties. We provide applications to TDLC completions of Baumslag-Solitar groups and certain groups of upper triangular matrices studied by Schesler.