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