Functional Analysis Seminar

In 1989, Benjamin Weiss proved that for a countable group G, there exists a global minimal G-flow which is a topological model for every essentially free p.m.p. Borel G-action. In this talk, I will present our vast generalization, asserting that any essentially free Borel measured action (without the p.m.p. assumption), admits a minimal compact model. Moreover, we show that given a random walk on the group, there exists a global minimal topological model for all essentially free stationary Borel actions.

Based on joint work with Yair Hartman.

Any countable group G comes equipped with a canonical dynamical system, namely its conjugation action on its space of subgroups Sub(G). This 0-dimensional compact space is a central object in measured and geometric group theory, especially because it supports invariant and stationary random subgroups.

In general, describing this space is hard. In 2024, Carderi, Gaboriau, Le Maître, and Stalder initiated the study of the space of subgroups of non-amenable Baumslag-Solitar groups BS(m,n). They provided an explicit description of the perfect kernel of Sub(BS(m,n)). This is the largest closed subspace without isolated points, i.e. the space that remains after performing successive derivations that remove the isolated points.

In this talk, I will provide a complete classification of the spaces Sub(BS(m,n)) up to homeomorphism. I will prove that there exist exactly four homeomorphism types among the non-amenable ones. This relies on a detailed study of the Cantor-Bendixson erasing process, which depends on the arithmetical properties of the parameters m,n. This is based on a joint work with Damien Gaboriau, François Le Maître, and Yves Stalder.

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