Timings: meeting at 14:00–17:30, followed by dinner.
Location: talks in B3.03, Zeeman Building.
14:00 Pénélope Azuelos (University of Bristol)
Narrow Schreier graphs and virtual fibre subgroups
A finitely generated subgroup H of a finitely generated group G is a virtual fibre subgroup if G admits a finite index subgroup which surjects onto the integers and the kernel has finite index in H. This condition is very strong; it implies many nice properties of the subgroup and imposes a number of geometric properties on the Schreier graph of H∖G. In this talk, I will discuss the extent to which various geometric properties of the Schreier graph, including the number of ends, growth and 'narrowness' characterise virtual fibre subgroups.
15:00 Luca Sabatini (University of Warwick)
Expanders, diameter bounds and group properties
Let G be a finite group and let X be a symmetric generating set. Which properties of Cay(G,X) genuinely depend on the choice of X? It is known that having very large diameter, i.e. greater than or equal to |G|^\epsilon, is in fact a group property. Answering a question of Pyber and Szabo, we exhibit a group G with two generating sets X and Y of fixed cardinality such that Cay(G,X) is an expander graph (in particular with diameter O(\log|G|)), while Cay(G,Y) has super-polylogarithmic diameter. This is joint work with S. Eberhard.
16:00 Tea & Coffee
16:30 Emmanuel Breuillard (University of Oxford)
Spectral gaps in finite groups of Lie type
Cayley graphs of finite simple groups of Lie type, as opposed to say abelian or nilpotent groups, are expected to be expander graphs. I will discuss progress towards a conjecture that in bounded rank, finite simple groups are uniformly expanding. This means that the spectral gap depends only on the rank and not on the Cayley graph, nor the group. The method relies on transferring to finite characteristic a uniformity result for Zariski-dense subgroups of complex semisimple algebraic groups proved by means of some ingredients from diophantine geometry. This is joint work with Oren Becker.
Timings: meeting at 13:00–16:30, followed by drinks and dinner.
Location: talks in Lecture Theatre A (G23), Watson Building (R15 on campus map)
13:00 Patricia Medina Capilla (University of Warwick)
Crown-based powers and their applications
Initially discovered by Gaschutz in the 1950s, before later being extended by Dalla Volta and Lucchini in the 1990s, the theory of crowns in finite groups has a long and rich history, with numerous applications. Central to this theory is the observation that establishing generation results for a particular class of groups, known as crown-based powers, is often sufficient to derive corresponding results for all finite groups. In this talk, we will explore how this framework can be applied to a range of generation problems, highlighting in particular how the structure of a group’s chief factors determines its generation behaviour.
14:00 Coen del Valle (Open University)
The binary actions of sporadic groups
An action of a finite group is called binary if the permutation group it induces has relational complexity 2. We will begin by discussing the concepts of relational complexity and binary actions of permutation groups, before exploring some recent progress towards classifying the binary actions of the finite simple groups, with a particular emphasis on the recently completed classification of binary actions of the sporadic groups.
15:00 Tea & Coffee
15:30 Martin Liebeck (Imperial College London)
p-exceptional permutation groups and applications to character theory
Let p be a prime. A p-exceptional permutation group is a transitive group G for which p divides the order of a point stabilizer H, but divides none of the orbit sizes of H. I will describe various classification results for p-exceptional groups, and show how they apply to questions in character theory, such as generalisations of the Gluck-Wolf theorem.