Geometric invariants of TDLC completions

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.