Abstract:

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, which have become a significant concept in topological group theory, as they establish a connection between abstract groups and TDLC groups. Recently, Bonn and Sauer investigated how the compactness properties behave under Schlichting completion. They showed that, from this point of view, 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 ongoing project, along with some of its applications.

[BS24] Laura Bonn and Roman Sauer. On homological properties of the Schlichting completion. Preprint, arXiv:2406.12740

[BHQ24a] Kai-Uwe Bux, Elisa Hartmann, and José Pedro Quintanilha. Geometric invariants of locally compact groups: the homological perspective. Preprint, arXiv:2411.13272

[BHQ24b] Kai-Uwe Bux, Elisa Hartmann, and José Pedro Quintanilha. Geometric invariants of locally compact groups: the homotopical perspective. Preprint, arXiv:2410.19501