| Résumé: | TBA |
| Résumé: | TBA |
| 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. |