Tuesday 1 October 2024, 14:00-15:00
Watson Building, B16
I'll give a brief introduction to the current state of the art regarding asymptotic dimension of box spaces (and possibly dynamic asymptotic dimension of certain group actions). While the question of what the dimension of these objects can be when it is finite has been more or less settled, it is still open whether box spaces of finite-dimensional amenable groups must have finite asymptotic dimension. I will discuss a possible approach to this question involving a connection between asymptotic dimension and Banach densities. In particular, I will show a lemma (an application of Kerr and Szabó's ε-quasitiling theorem) which says, speaking somewhat loosely, that a box space of an amenable group has the same asymptotic dimension as the group except possibly on a subset with vanishing upper density (measure zero).
Tuesday 8 October 2024, 14:00-15:00
Watson Building, B16
The general linear group GLn(𝕜) over a field 𝕜 acts on the space 𝔤𝔩n(𝕜) of (n × n)-matrices by conjugation. The set 𝒩(𝔤𝔩n(𝕜)) of nilpotent matrices is preserved by this action and the group acts with finitely many orbits, with each orbit having a representative given by the Jordan normal form that is independent of 𝕜. In fact, more is true. The structure of the stabiliser of this nilpotent matrix is even independent of 𝕜.
The natural setting for these questions is the setting of connected reductive algebraic groups. One can ask to what extent these types of results hold for arbitrary split connected reductive groups. In this talk I will outline several reasonable ways to generalise the situation of GLn, both by changing GLn(𝕜) and the space 𝔤𝔩n(𝕜) on which it acts.
This is on-going joint work with Adam Thomas (Warwick).
Tuesday 15 October 2024, 14:00-15:00
Watson Building, B16
A group G is said to be free-by-cyclic if it contains a normal subgroup N which is free of finite rank and such that the quotient of G by N is infinite cyclic. Free-by-cyclic groups form a large class of groups which exhibits a range of interesting behaviour. A group G is said to be equationally Noetherian if any system of equations over G is equivalent to a finite subsystem. In joint work with Motiejus Valiunas, we show that all free-by-cyclic groups are equationally Noetherian. As an application, we deduce that the set of exponential growth rates of a free-by-cyclic group is well ordered.
Tuesday 22 October 2024, 14:00-15:00
Watson Building, B16
In general, it is hard to characterise "all subgroups" of a given group -- even hyperbolic groups still have many mysteries here. However, restricting the complexity in some way can make the problem tractable: subgroups of free groups, or of surface groups are not so bad, and cyclic subgroups don't cause too many problems. Two generators is a lot more than one, but progress can still be made: in 1979, Jaco and Shalen characterised the two-generator subgroups of fundamental groups of certain orientable three manifolds.
I will talk about recent work with Edgar Bering, Ilya Kapovich and Stefano Vidussi characterising the two-generator subgroups of mapping tori of free groups, using ideas from Feighn and Handel's proof of coherence for these groups.
Tuesday 5 November 2024, 14:00-15:00
Watson Building, B16
CW-complexes are topological spaces built inductively by gluing cells of different dimensions in specific ways. Given a group, one can canonically associate it with an aspherical CW complex, namely its classifying space. This gives us an equivalence between the category of groups and homomorphisms and the category of (aspherical) CW-complexes and cellular maps. We will try to better understand this dictionary through the lens of finiteness properties Fn, FPn and type F. I will then define the group cohomology and cohomological dimension. We will see how the cohomological dimension of a group G relates to the geometric dimension of G - the smallest possible dimension of a classifying space for G. A Z-structure on a group consists of a CW-complex X on which G acts freely and cocompactly and a 'nice' compactification G̅ of X. The topology of the "boundary" Z=X̅-X carries algebraic information about the group G. In particular, the topological dimension of Z equals the cohomological dimension of G. I will end by reviewing some known results and open questions about Z-structures.
Tuesday 12 November 2024, 14:00-15:00
Watson Building, B16
We call a group and a conjugacy class of involutions a k-transposition group if the class generates the group and any product of two elements in the class have order less than or equal to k. In 1970, Fischer investigated 3-transposition groups and as a result found three new sporadic simple groups. He also predicted the Baby Monster and Monster simple groups by looking at larger transposition groups. Since then, others have looked at certain transposition groups and produced classifications. In this talk, we will start by discussing the history before giving basic observations and results. The rest of the talk will be focused on the progress so far on simple, almost simple, and quasisimple groups and figuring out when they are or not 6-transposition groups. This is joint work with Chris Parker.
Tuesday 19 November 2024, 14:00-15:00
Watson Building, B16
Let n be a sufficiently large positive integer. A character is multiplicity-free if its irreducible constituents appear with multiplicity one. Wildon in 2009 and independently Godsil and Meagher in 2010 have found all multiplicity-free permutation characters of the symmetric group Sn. In this talk, we focus on a significantly more general problem when the permutation characters are replaced by induced characters of the form ρ↑Sn with ρ irreducible.
Despite the nature of the problem, I explain, combining results from group theory, representation theory and combinatorics, why this problem may be feasible and present a close to full answer. I also mention some of my (often surprising) results to questions about conjugate partitions, which naturally arise when solving the problem, and the remarkable complete classification of subgroups G of Sn, which have an irreducible character which stays multiplicity-free when induced to Sn.
Tuesday 26 November 2024, 14:00-15:00
Watson Building, B16
For a natural number n and p a prime, let Pn denote a Sylow p-subgroup of the symmetric group Sn. In 2017 E. Giannelli and G. Navarro proved that if χ is an irreducible character of Sn with degree divisible by p, then the restriction of χ to Pn has at least p different linear constituents. In this talk, we classify which characters of Sn whose restriction to Pn, has exactly p distinct linear constituents (not counting multiplicity). Most of the talk will focus on the p = 2 case. For odd primes, the classification follows more directly from results by S. Law and E. Giannelli.
This is joint project with Stacey Law.
Tuesday 10 December 2024, 14:00-15:00
Watson Building, B16
In this talk, we give a brief overview of s-arc-transitive graphs and show how their study in the case of cubic (3-regular) graphs reduces to a generation problem for finite almost simple groups. We then discuss current progress towards solving this generation problem for PSL(n,q) when n is sufficiently large and q is prime to 6.