20th Meeting: Groups & Geometry

Birkbeck, University of London, Wednesday December 11th 2019

Venue: room G04 of 43 Gordon Square. On entering the building walk towards the coffee shop in front of you and immediately take a right. Confusingly, the crazy logic of the room numbering involves having to walk past G03 and G05 to get to G04. This is the same building as the past couple of years. (You’re warned that an accurate map of the building itself can be found here.) If you would like some directions then email b.fairbairn[usual funny AT symbol]bbk.ac.uk. Afterwards we will very likely go for some refreshments nearby and a curry afterwards.

13:30-14:30 Matthew Conder (Cambridge) Discrete and free two-generated subgroups of SL2 over non-archimedean local fields

It is well known that the Ping Pong Lemma can be applied to many two-generated subgroups of SL(2,ℝ) (using the action by Möbius transformations on the hyperbolic plane) in order to determine properties such as freeness and/or discreteness. In particular, there is a practical algorithm (of Eick, Kirschmer and Leedham-Green) which, given any two elements of SL(2,ℝ), will determine after finitely many steps whether or not the subgroup generated by these elements is both discrete and free of rank two. In this talk, I will show that a similar algorithm exists for two-generated subgroups of SL(2,K), where K is a non-archimedean local field (for instance, the p-adic numbers). Such groups act by isometries on a Bruhat-Tits tree, and the algorithm proceeds by computing and comparing various translation lengths, in order to determine whether or not a given two-generated subgroup of SL(2,K) is both discrete and free.

14:30-15:30 Brita Nucinkis (RHUL) An irrational slope Thompson’s group

In this talk I will discuss a relative to Thompson’s group F, the group Fτ, which is the group of piecewise linear homeomorphisms of [0,1] with breakpoints in ℤ[τ] and slopes powers of τ, where τ = (√5 -1)/2 is the small Golden Ratio. This group was first considered by S. Cleary, who showed that the group was finitely presented and of type F. Here we take a combinatorial approach considering elements as tree-pair diagrams, where the trees are finite binary trees, but with two different kinds of carets. We use this representation to show that the commutator subgroup is simple and give a unique normal form for its elements. The surprising feature is that the T – and V – versions of these groups are not simple, however. This is joint work with J. Burillo and L. Reeves.

15:30-16:00 Break

16:00-17:00 Martin Bridson (Oxford) Direct products inside the automorphism groups of free groups

There is a bound, 2n-3, on the maximal number of factors in a direct product of non-elementary (equivalently, free) subgroups inside Aut(Fn), the automorphism group of a free group of rank n. I shall explain how we establish this bound, and explain the classification of the maximal products. The proof relies on a refined understanding of the action of Aut(Fn) on spaces of Fn-trees, and on the structure of auxiliary hyperbolic complexes. If time allows, I will also discuss applications to commensurator rigidity. (This is joint work with Ric Wade, Oxford.)

19th Meeting: Infinite Groups

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.

18th Meeting: Finite Groups

University of Manchester, Wednesday April 3rd 2019, room G108 of the Alan Turing building

Owing to exceptional budget constraints we politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine – note that most people in this category are elgibile for a rail-card).

13:00-13:50 Awatef Almotairi (Manchester) Induced Actions of Permutation Groups on k-subsets

A permutation group acting on a set of size n induces an action on the set of all k-subset where k<n. This talk will be concerned with various properties of this induced action and will report on recent results on what can be said about orbit lenghts.

14:00-14:50 Martin van Beek (Birmingham) Fusion Systems of Characteristic p-type

We present a fusion theoretic analogue of groups of characteristic p-type and, following a paper of Michael Aschbacher in 2009 who examined fusion systems of characteristic 2-type, we discuss generation properties of these fusion systems, for p an odd prime. The results are similar in spirit to a ‘global’ C*(G,T)-theorem for groups given by Gyde Autzen in 2009.

15:00-15:45 Break

15:45-16:35 Rob Wilson (QMUL) Algebra in a vacuum

The symmetry group of empty space (aka the vacuum) is SO(3) in Newtonian physics, SO(3,1) in Special Relativity, and SL(4,R) in General Relativity. Properties of the vacuum therefore correspond to properties of the representation theory
of these groups. Quantum mechanics is based on a double cover of SO(3,1),
as well as the so-called gauge groups U(1), SU(2) and SU(3). The big
question in fundamental physics is, how do these groups relate to each other?
Guided by the experimental evidence, I search the group theory and the
representation theory for clues.

17th Meeting: Groups & Representations

University of Birmingham, Wednesday February 27th 2019

All talks will take place in the Watson building. The first two talks will be in room LRC and the last talk will be in room LRA.

Owing to exceptional budget constraints we politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine).

13:30-14:30 Imen Belmokhtar (QMUL) The Structure of Induced Simple Modules for 0-Hecke Algebras

We shall be concerned with the 0-Hecke algebra; its irreducible representations were classified and shown to be one-dimensional by Norton in 1979. The structure of a finite-dimensional module can be fully described by computing its submodule lattice. We will discuss how this can be encoded in a generally much smaller poset given certain conditions; this allows us to obtain branching rules which remarkably describe the full structure of an induced simple module in types B and D.

14:30-15:30 Florian Eisele (Glasgow) Picard groups of blocks and Donovan’s conjecture

While the outer automorphism group of a block over an algebraically closed field of characteristic p is usually an infinite group, it was recently observed by Boltje, Kessar and Linckelmann that the outer automorphism group of a block defined over a suitable discrete valuation ring is finite in all known examples.

As of yet there is no general explanation for the finiteness of these outer automorphism groups, and the closely related Picard groups of blocks. I will talk about recent results on the structure of these groups, and an application to Donovan’s conjecture (which is joint work with C. Eaton and M. Livesey).

15:30-16:15 Break

16:15-17:15 Gabriel Navarro (Valencia) Why the Galois-McKay conjecture and new consequences

While the celebrated McKay conjecture asserts that two numbers, one local, the other global, are in fact the same, the Galois version of this conjecture is strong enough to relate certain global and local structures.

We will give a survey on this and offer new recent consequences.

16th Meeting: Groups & Geometry

Birkbeck, University of London, Wednesday December 12th 2018

Venue: Room B03 of 43 Gordon Square. This is the same building and even the same floor of that building as last year’s room, however is not the same room. You’re warned that an accurate map of the building itself can be found here.) If you would like some directions then email b.fairbairn[usual funny AT symbol]bbk.ac.uk. Afterwards we will very likely go for some refreshments nearby and a curry afterwards.

We politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine).

13:30-14:30 Motiejus Valiunas (Southampton) Acylindrical hyperbolicity of graph products

Graph products are a class of groups that interpolate between direct and free products, and generalise the notion of right-angled Artin groups. It is known that most graph products belong to the class of acylindrically hyperbolic groups, which also includes and shares many properties with (relatively) hyperbolic groups. A recent result suggests a relation between this class and the class of equationally noetherian groups: these are groups for which any system of equations has the same solution set as some finite subsystem.

In this talk I will explicitly describe an acylindrical action of a graph product on a quasi-tree, which gives an alternative way to see acylindrical hyperbolicity of these groups. As an application, I will discuss how this action can be used to show that, under certain conditions, the property of being equationally noetherian is preserved under forming graph products.

14:30-15:30 Ian Leary (Southampton) Subgroups of almost finitely presented groups

In 1949 Higman-Neumann-Neumann showed that every countable group embeds in a 2-generator group. In 1961 Higman described which finitely generated groups embed in finitely presented groups. The class of almost finitely presented groups
lies somewhere between ‘finitely generated’ and ‘finitely presented’.  I will describe these classes, and my recent analogue of the HNN and Higman embedding theorems

15:30-16:00 Break

16:00-17:00 Aditi Kar (Royal Holloway) 2D Problems in Groups

In this talk, I will describe how Nikolov and I connect a naive conjecture of ours about stabilisation of deficiency over finite index subgroups with Wall’s D2 Problem and the Relation Gap problem. Wall’s Problem is classical and dates back to a paper of Wall’s from the 1950’s; in a way, the Relation Gap Problem is folklore. I will describe a solution to the pro-p version of the conjecture, as well as its higher dimensional abstract analogues.

15th Meeting: Finite Groups

University of Manchester, Wednesday May 23rd 2018

Venue: Room G108 of the Alan Turing Building, University of Manchester.

Owing to exceptional budget constraints we politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine – note that most people in this category are elgibile for a rail-card).

13:00-13:50 Raul Moragues Moncho (University of Birmingham) Fusion systems over p-groups with an extraspecial subgroup of index p

Saturated fusion systems are constructions on p-groups which generalise the action of a finite group on its Sylow p-subgroups via conjugation. They have uses in representation theory and topology, but we focus on finite simple groups. Starting from a p-subgroup, we introduce homomorphisms between its subgroups which behave like conjugation by an element of an overgroup. There arise exotic configurations which do not come from finite groups, one of which was found in the search for the finite simple groups.

In order to find and study new exotic configurations, fusion systems have been classified over certain families of p-groups. We focus on p-groups S with an extraspecial subgroup of index p when p is odd, which appear as Sylow p-subgroups of PSL4(p), PSU4(p), PSp4(p) and G2(p) if p is large enough, as well as some other finite simple groups when p is small. We present a classification of fusion systems over S with no normal p-subgroups, which includes some new exotic examples related to a family constructed by Parker and Stroth.

14:00-14:50 Scott Harper (University of Bristol) 3/2-Generation of Finite Groups

It is well known that every finite simple group can be generated by two elements. Moreover, two arbitrary elements are very likely to generate the entire group. For example, every non-identity element of a finite simple group belongs to a generating pair. Groups with the latter property are said to be 3/2-generated. It is natural to ask which other finite groups are 3/2-generated. In 2008, Breuer, Guralnick and Kantor conjectured that a finite group is 3/2-generated if and only if every proper quotient of the group is cyclic.

In this talk, I will introduce this area of research before reporting on recent progress towards establishing this conjecture, where probabilistic techniques play a key role. I will also highlight connections with related topics, such as recent work on generating graphs.

15:00-15:45 Break

15:45-16:35 Sarah Rees (University of Newcastle) Negative curvature in some Artin groups

Artin groups are those groups with presentations of the form
< x1, x2, …, xn | xixjxi…= xjxixj… where each word has length mij, 0<i,j<n+1>
where mij are at least 2 and possibly infinite. Unless an Artin group is free, it
can’t be hyperbolic because it contains subgroups isomorphic to the free abelian group of rank 2. But despite  that, it’s been known for more than 30 years that many Artin groups (certainly all those of ‘large’ type, for which no pairs of generators commute, or rather,  mij>2, for all i,j) exhibit a weak form of relative hyperbolicity. Recent work of Huang and Osajda now proves that all those Artin groups, and a few more, are ‘systolic’, that is, they act geometrically on ‘systolic complexes’. Hence, in particular, they are all biautomatic, and display various algorithmic properties which are always found in hyperbolic groups.

This is consistent with my own relatively recent work with Holt, which proved that all these Artin groups (and more) are shortlex automatic, with effective rewrite systems to a (shortlex minimal) normal form. I’ll talk about ongoing work with Holt, to try and explain how these quite different approaches give many of the same answers.

14th meeting: Finite Groups

University of Bristol, Wednesday February 7th 2018

Venue: Lecture theatre SM1, School of Mathematics, University of Bristol

Owing to exceptional budget constraints we politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine – note that most people in this category are elgibile for a rail-card).

14:00-15:00 Carlisle King (Imperial College, London) Probabilistic methods and generation of finite simple groups

Let G be a finite group. It is well-known that G is generated by a pair of elements – we say G is 2-generated. Later, it was shown that 2 elements chosen at random from G generate G with probability tending to 1 as |G| tends to infinity. A natural refinement is to ask, given a pair of integers (a,b), whether G is 2-generated by an element of order a and an element of order b. If such a pair exists, we say that G is (a,b)-generated. We will explore some past results regarding (2,3)-generation proved using probabilistic methods, as well as a recent result on (2,p)-generation for some prime p.

An equivalent statement of the 2-generation theorem is that every finite simple is an image of F2, the free group on 2 generators. More generally, given a finitely presented group Γ, one can ask which finite simple groups are images of Γ. We will study a new result in this area concerning the free produc Γ=A*B of nontrivial finite groups A and B, also proved using probability methods.

15:00-15:30 Break

15:30-16:30 Stacey Law (Cambridge) p’-branching for symmetric groups

Let p be any prime and n be any natural number. Let χ be an ordinary irreducible character of the symmetric group Sn whose degree is coprime to p.

We bound the number of p’-irreducible constituents of the restriction of χ to Sn−1. This generalizes work of Ayyer, Prasad and Spallone (2016) for the prime p = 2. This is joint work with E. Giannelli and S. Martin.

16:30-17:30 Nick Gill (South Wales) Cherlin’s conjecture, Lachlan’s theory of homogeneous relational structures and the notion of “sporadicness”

Some years ago, Lachlan advanced a theory on homogeneity in relational structures which imposed a natural “hierarchy of complexity” on the universe of homogeneous relational structure. This theory was reworked by Cherlin in the 1990’s with a view to understanding finite permutation groups from a model theoretic point of view.

One upshot of all this is that we know the existence of an infinite family of theorems describing the so-called “relational complexity”  of all finite permutation groups. The problem is that, although we know these theorems exist, and even have a “form” for them, nonetheless we do not yet have the precise statement of any of them. However Cherlin has conjectured what (part of) the first of these theorems should say, and we will discuss this conjecture at some length.

There has also been substantial progress on this conjecture due to Cherlin himself, to Wiscons, and to myself and various co-authors. In particular the recent results that I will describe are due to myself, Pablo Spiga, Francis Hunt and Francesca Dalla Volta.

The talk has elements of model theory, combinatorics and finite permutation groups, and should be accessible to all.