University of Bristol, Wednesday May 29th 2019

SM1, School of Mathematics, University of Bristol

13:30-14:30 Damian Sercombe (Imperial College London) Classifying maximal *k*-subgroups of simple algebraic *k*-groups

Let *k* be an arbitrary field. We are interested in classifying the maximal *k*-subgroups *H* of any simple (linear) algebraic *k*-group *G* up to conjugacy by some element of *G*(*k*). We call this the classification problem. The subproblem where *H* is *k*-anisotropic is called the anisotropic classification problem – it appears to depend intrinsically on the choice of field *k*. The classification problem has been solved when *k* is algebraically closed, *k* is finite and *k*=R. However, very little is known for other fields.

We develop techniques based on Galois cohomology and Borel and Tits’ theory of algebraic *k*-groups to reduce the classification problem to the anisotropic classification problem in full generality (i.e. independently of the choice of *k*). In particular, we utilise an important invariant of a connected reductive algebraic *k*-group called its index that was introduced by Satake and Tits. We apply these techniques to the cases where *G* is an exceptional *k*-group and *H* is a *k*-isotropic maximal connected *k*-subgroup of maximal rank in *G*. If *k* is algebraically closed, *k* is finite or *k*=R, then our classification reduces to what is known in the literature.

14:30-15:30 Nikolay Nikolov (Oxford) On conjugacy classes in compact groups

It’s a classical result that the number of conjugacy classes of a finite group *G* tends to infinity as |*G*| tends to infinity and that this is not true if *G* is infinite. In this talk I will present joint work with Andrei Jaikin-Zapirain. Our result is that an infinite Hausdorff compact group has uncountably many conjugacy classes. The proof relies on the classification of finite simple groups and the study of finite groups with almost regular automorphisms.

15:30-16:15 Break

16:15-17:15 Christopher Voll (Bielefeld) Normal subgroup growth of free nilpotent groups under base extension – wild or uniform?

The normal subgroup growth of a group captures the asymptotic behaviour of the lattice of its normal subgroups of finite index. The group’s normal zeta function encodes the distribution of these subgroups quantitatively.

Finitely generated nilpotent groups are arithmetic subgroups of unipotent algebraic groups. It therefore comes as no surprise that questions about their subgroup structure are best thought about, and answered, in number-theoretic and, occasionally, algebro-geometric terms.

I will explain a notion of uniformity for normal zeta functions, specifically for free nilpotent groups, and present a recent uniformity result for free nilpotent groups of class 2, answering a conjecture of Grunewald, Segal, and Smith in this case.

My talk is based on joint work with Angela Carnevale and Michael Schein.