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 TBA (TBA)

TBA

14:30-15:30 TBA (TBA)

TBA

15:30-16:15 Break

16:15-17:15 TBA (TBA)

TBA

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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.

13th meeting: Groups & Geometry

SUBJECT TO THE LMS AWARDING FUNDING TO THE “TRIANGLE” NEXT YEAR…

Birkbeck, University of London, Wednesday December 13th 2017

Room B13 of 43 Gordon Square. Note that this is not the usual building but is nearby (if you’ve walked to Birkbeck from somewhere like Euston or King’s Cross before then you’ve practically walked past it. 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 The Bree Louise followed by a curry (assuming they’re still standing – like Brexit and Winter, HS2 is coming…)

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 Emilio Pierro (LSE) Big groups and their small quotients

In this talk we will discuss new techniques in determining which finite simple groups can and cannot occur as quotients of certain infinite groups. Our motivation is to prove a conjecture of Mechia-Zimmerman stating that Ln(2) is the smallest non-trivial quotient of Aut(Fn), the automorphism groups of the free group of rank n. Our techniques can also then be applied to solve the analogous conjecture of Zimmerman that S2g(2) is the smallest non-trivial quotient of MCG( Σg,b), the mapping class group of a connected, orientable surface of genus g>2 with b non-negative boundary components and no punctures. This is joint work with Barbara Baumeister and Dawid Kielak.

14:30-15:30 Henry Bradford (Göttingen) Short Laws for Finite Groups and Residual Finiteness Growth

A law for a group G is a non-trivial equation satisfied by all tuples of elements in G. We study the length of the shortest law holding in a finite group. We produce new short laws holding (a) in finite simple groups of Lie type and (b) simultaneously in all finite groups of small order. As an application of the latter we obtain a new lower bound on the residual finiteness growth of free groups. This talk is based on joint work with Andreas Thom.

15:30-16:00 Break

16:00-17:00 Caroline Series (Warwick) Geometry in non-discrete groups of hyperbolic isometries: Primitive stability and the Bowditch condition are equivalent

It appears that there can be geometrical conditions on a group of hyperbolic isometries which may sometimes be of interest even when the group is not discrete.  We explain two different such conditions which pertain to the primitive elements in an SL(2,C) representation of the free group F2.  One is Minsky’s condition of primitive stability, and the other is the so-called BQ-condition introduced by Bowditch and generalised by Tan, Wong and Zhang. We prove these two conditions are equivalent, aided by an auxiliary condition which constrains the  location of the axes of those primitive elements which are palindromic words.

12th meeting: Permutation Groups

University of Bristol, May 17th 2017

School of Mathematics, University Walk, lectures will take place in room SM2.

After the last talk we will go for drinks and dinner.

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).

14:00 – 15:00 Joanna Fawcett (University of Cambridge) Finite k-connected-homogeneous graphs

A graph is homogeneous if any isomorphism between finite induced subgraphs extends to an automorphism of the entire graph. The finite homogeneous graphs have been completely classified, and only a few families of examples arise. In this talk, we will discuss several methods of relaxing the hypothesis of homogeneity. In particular, we will discuss some recent progress on classifying the finite k-connected-homogeneous graphs where k>3. This is joint work with A. Devillers, C.H. Li, C.E. Praeger and J.-X. Zhou.

15:00 – 15:30 Break

15:30 – 16:30 Melissa Lee (Imperial College) Bases of quasisimple linear groups and Pyber’s conjecture

A base of a group G acting faithfully on a set  Ω is a subset B⊆Ω such that the pointwise stabiliser of B in G is trivial. The minimal base size of G is denoted by b(G).

A well-known conjecture made by Pyber in 1993 states that there is an absolute constant c such that if G acts primitively on Ω, then b(G) < clog|G|/log n, where |Ω|=n.

Following the contributions of several authors, the conjecture was finally established in 2016 by Duyan, Halasi and Maróti.

A result that played a major role in the proof of Pyber’s conjecture for primitive linear groups was given by Liebeck and Shalev, who proved that there is a constant C such that if G is a quasisimple group acting irreducibly on a finite vector space V, then either b(G)C, or G is an alternating or classical group acting on its natural module. In this talk, I will cover the history of Pyber’s conjecture, especially in the context of primitive linear groups, and present some results on the determination of the constant C for bases of quasisimple groups. I will also discuss an application by Liebeck of the latter result, which improves the known upper bounds for b(G) when G is an irreducible primitive linear group.

16:30 – 17:30 Simon Smith (University of Lincoln) The box product of two permutation groups

There are a number of ways in which one may take the product of two groups. Products which possess some kind of “universal” property (like the free and wreath products), or those which preserve some of the important properties of the input groups, are rare and precious.

Arguably, the most important product in permutation group theory is the wreath product, acting in its product action. The reason for this is that, unlike other products, it preserves a fundamental property called primitivity. Primitive permutation groups are indecomposable in some sense, and for finite groups they are the basic building blocks from which all permutation groups are comprised.

I am going to talk about a new product, called the box product. It is fundamentally different to the wreath product in product action. Nevertheless, it preserves primitivity under astonishingly similar conditions. Moreover, the box product has a “universal” property, and under natural conditions on groups M and N, the product of M and N is simple. The product can be used to easily solve a well-known open problem from topological group theory, and has an important role to play in the classification of infinite permutation groups.

11th meeting: Finite Groups

University of Manchester, Wednesday March 1st 2017

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).

All talks will be held in Room G205 of the Alan Turing Building

1.30 – 2.20 Sarah Hart (Birkbeck)  Product-free sets and Filled Groups

2.30 – 3.20 Ali Aubad (Manchester) Commuting Involution Graphs for Double Covers of the Symmetric Groups

3.30 – 4.00   Refreshments

4.00-4.50 David Ward (Manchester) Cuspidal Character and Finite Sporadic Simple Groups

The last talk will be followed by going for further refreshment followed by a curry.