Let $\Gamma$ be a Hausdorff topological group and $\Lambda$ an open commensurated subgroup of $\Gamma$. A TDLC completion (short for totally disconnected locally compact completion) of $(\Gamma,\Lambda)$ is a pair $(G,\phi)$ where: $G$ is a TDLC group, $\phi\colon \Gamma\to G$ is a continuous homomorphism with dense image and $\Lambda=\phi^{-1}(L)$ for some compact open subgroup $L\subseteq G$. One important example is given by Schlichting completions.
 
Recently, Bonn and Sauer showed that, from the point of view of compactness properties, the Schlichting completion of a pair $(\Gamma,\Lambda)$ acts precisely as if it were the quotient of $\Gamma$ by $\Lambda$ [BS24]. Motivated by this result, José Pedro Quintanilha and I set out to explore whether a similar phenomenon holds for the $\Sigma$-sets [BHQ24a,BHQ24b].
 
In this talk, I will present the results of our project [CQ25], along with some of its applications.