| Résumé: | Right-angled Artin groups (RAAGs) are a natural class of examples in geometric group theory. These groups have a finite presentation where every relator is the commutator of two generators, which can be encoded by a graph. A result of Droms states that two RAAGs are isomorphic if and only if their defining graphs are isomorphic. However, deciding whether one RAAG embeds as a subgroup of another is a much harder problem. Substantial progress was made by Kim and Koberda, notably solving the two-dimensional case, relying on their definition of the extension graph of a RAAG, an object somewhat analogous to the curve graph of a surface. Quasi-isometric embedding between RAAGs was also recently studied by Bader–Bensaïd–Petyt. I will present a joint work with Sasha Gurieva and Camille Horbez where we investigate measurable embedding, a measured analogue of coarse embedding. We completely characterize measurable embedding among square-free RAAGs (i.e. RAAGs that do not contain F2 x F2). Interestingly, the characterization is both in terms of subgroup embedding and of extension graphs. It is also algorithmically decidable. |
| Résumé: | Any countable group $G$ comes equipped with a canonical dynamical system, namely its conjugation action on its space of subgroups $\mathrm{Sub}(G)$. 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 $\mathrm{BS}(m,n)$. They provided an explicit description of the perfect kernel of $\mathrm{Sub}(\mathrm{BS}(m,n))$. 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 $\mathrm{Sub}(\mathrm{BS}(m,n))$ 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. |
| Résumé: |
Given a finitely generated discrete group, one can define a natural topology on it by declaring finite-index subgroups to form an open neighbourhood of the identity; this is called the profinite topology. Then, one is interested in which subgroups of the group are closed in the profinite topology. This leads to the notions of residually finite, LERF or ERF, depending on whether the trivial subgroup, all finitely generated subgroups or all subgroups are closed in the profinite topology respectively. Residually finite groups are precisely those acting on rooted spherically homogeneous rooted trees, while the LERF and ERF properties are much more mysterious in general. A classical result of Hall shows that free groups are LERF, which was later extended to limit groups by Wilton. Examples of ERF groups are difficult to find, and they have only been found among virtually polycyclic groups.
We will be interested in two well-known classes of groups acting on rooted trees: branch and weakly branch groups. These classes of groups have played a prominent role in geometric group theory by exhibiting examples of groups of intermediate growth, amenable but not elementary amenable groups, and strong links to complex and arithmetic dynamics. The LERF property has been studied for branch groups since the early 2000s, but currently only very ad hoc methods are available for the study of this property. Thus, the amount of examples of LERF branch groups is extremely limited. In this talk, we shall focus on the ERF property instead and see what can be said about it for branch and weakly branch groups. This is joint work with Paul-Henry Leemann and Tatiana Nagnibeda. |