Schedule

In this talk I will present recent work with Samuel Corson, Phillip Möller and Olga Varghese on the question of profinite rigidity of Coxeter groups.  Here a group is profinitely rigid if it is determined up to isomorphism by its finite quotients amongst all finitely generated residually finite groups.  I will present two results, first that there are right-angled Coxeter groups (RACGs) which have infinitely many non-isomorphic subgroups with the same finite quotients, and second, that amongst Coxeter groups, RACGs are determined by their finite quotients.

( joint work with J. Delgado and M. Roy)  For two subgroups H_1, H_2 of a group G, we can look at the eight possibilities for the finite/non-finite generability of H_1, H_2, and H_1 intersected H_2. For example, all eight are possible in a free non-abelian group except one of them, expressing the well-known fact that free groups are Howson: intersection of two finitely generated subgroups is again finitely generated. A group G is called intersection-saturated when, for every k, each of the 2^(2^k-1) such k-configurations is realizable by appropriate subgroups H_1,... ,H_k of G.

In this talk, we prove the existence of explicit finitely presented intersection-saturated groups. We also show that the Howson property is the only restriction for realizability in free groups: a k-configuration is realizable in a free non-abelian group if and only if it respects the Howson property.

A group is just infinite if it is infinite but all of its proper quotients are finite. Branch groups form one of the three possible classes into which just infinite groups can fall, according to Wilson's classification. They also often have unusual algebraic and geometric properties, such as periodicity and intermediate growth, which makes them quite intriguing. In this talk, we will focus our attention on the question of determining which finitely generated groups embed into a given just infinite branch group, and the way in which they can be embedded. We will present new results that will allow us to obtain very precise answers for groups possessing a property known as the Subgroup Induction Property. This is joint work with Rostislav Grigorchuk, Paul-Henry Leemann and Tatiana Nagnibeda.

Self-replicating groups are isometry groups of rooted trees which furnish important examples of groups in geometric and combinatorial group theory. Scale groups are isometry groups of regular trees which arise as subquotients of general totally disconnected, locally compact (t.d.l.c.) groups.

It will be explained how these classes of groups are essentially equivalent despite acting on different types of trees. This equivalence allows an exchange of ideas between the study of discrete groups on one hand and t.d.l.c. groups on the other. In particular, the role of scale groups in the structure theory of general t.d.l.c. groups raises general questions about the structure of scale groups.

In 1973, William Boone and Graham Higman proved that a finitely generated group G has a solvable word problem if and only if G can be embedded into a simple subgroup of a finitely presented group. They conjectured a stronger result, namely that every such group G embeds into a finitely presented simple group. This conjecture remains open after almost 50 years, but recent advances in the study of finitely presented simple groups have made it possible to verify the Boone-Higman conjecture for several large classes of groups. In this talk, I will survey recent progress on Boone-Higman embeddings of right-angled Artin groups, countable abelian groups, contracting self-similar groups, and hyperbolic groups.

This talk includes joint work with Jim Belk, Collin Bleak, James Hyde, and Matthew Zaremsky. 

In this talk, I will present some new results on Dade's Conjecture for finite reductive groups. If p is a prime number dividing the order of a finite group G, then Dade's Conjecture gives a formula to count the number of irreducible representations in a Brauer block of G and with a given p-defect in terms of the p-local structure of the group. This behavior is in accordance with a recurring phenomenon known as the local-global principle in representation theory of finite groups. When G is a finite reductive group arising as the set of rational points of a linear algebraic group and p is different from the defining characteristic, the p-local structure of G can be replaced by a geometric analog: the e-local structure. Using this observation I will explain how Dade's Conjecture can be explained within the framework of generalised Harish-Chandra theory.

The finite p-groups of fixed coclass r can be sorted into a graph G(p,r). Die vertices of this graph correspond to the isomorphism types of p-groups of coclass r and there is an edge G -> H if G is isomorphic to the class-c quotient of H, where H has class c+1. The graphs G(p,r) reveal a number of interesting periodic patterns. Some of these patterns have been proved, others are conjectured based on extensive computational evidence. This talk surveys the state of the art of this research area.