| Résumé: |
Maximal subgroups play a very important role in our understanding of finite groups. In the setting of infinite groups maximal subgroups remain rather mysterious objects, despite a lot of research. A good place to start is the seminal theorem of Margulis and Soifer.
Theorem: A finitely generated linear group, admits a maximal subgroup of infinite index if and only if it is not virtually solvable. Since its publications in the 1970's this theorem inspired a great deal of research. I will describe the main ideas involved in the proof of the theorem and then survey the many ramifications and developments that followed. My talk will be based on a a recent survey paper written by Gelander, Soifer and myself on the occasion of Margulis' birthday. |
| Résumé: | A contraction group is a pair $(G,\tau)$, where $G$ is a locally compact group and $\tau$ is an automorphism of $G$ such that $\tau^n(g)\to e$ as $n\to\infty$ for all $g\in G$. It follows from results in H. Glöckner and G. A. Willis, J. Reine Angew. Math., 634 (2010), 141–169 that, if $G$ is a totally disconnected contraction group and is locally pro-$p$, then it is the direct product of a $p$-adic Lie group and a torsion group having a composition series in which each composition factor is isomorphic to the additive group of $\mathbb{F}_p(\!(t)\!)$. It has long been known that $p$-adic Lie contraction groups are nilpotent. These ideas and results will be reviewed in the talk. Then a proof will be sketched that the torsion factor is nilpotent too. It follows, therefore, that every locally pro-$p$ contraction group is nilpotent. This is joint work with H. Glöckner. |
| Résumé: | We develop a generalisation of Burger-Mozes universal groups, acting on the $d$-regular tree $T_{d}$, by prescribing the local action on vertex neighbourhoods of a given radius $k$. In the locally transitive case, these groups are precisely the Banks-Elder-Willis $k$- closures of subgroups of $\mathrm{Aut}(T_{d})$ which contain an edge inversion of order $2$. As an application, we show that the simple groups associated to certain groups with pairwise distinct $k$-closures are pairwise non-isomorphic as topological groups. |
| Résumé: | The transitivity degree of a group $G$ is an invariant of $G$ associated to representations of $G$ as a permutation group: this is the largest integer $k$ such that $G$ admits a faithful $k$-transitive action; and if there is no such integer then the transitivity degree is infinite. For several classes of groups coming from geometric group theory, this invariant is known to be infinite, notably by a result of Hull–Osin. Recently Gelander–Glasner–Soifer showed that the same holds for Zariski dense subgroups of $\mathrm{SL}(2,K)$, where $K$ is a local field. In the talk we will discuss the situation of certain groups coming from "dynamical group theory". For example for groups $G$ having a minimal and non-free action on the circle by homeomorphisms, we will see that the transitivity degree can be explicitly computed, and is determined by the action of $G$ on its orbits on the circle. This is joint work with Nicolas Matte Bon. |