LMS Midlands Regional Meeting & Workshop 2025
Groups in the Midlands
17-20 June 2025, Lincoln, UK
Schedule
The Society Meeting
starts on Tuesday (17 June) at 13:00, with Greetings and Society Business. The
last talk ends at 17:15, followed by a Reception. The Society Dinner starts at 18:30 at Cosy Club Lincoln.
The Workshop "Groups in the Midlands"
starts on Wednesday (18 June) at 10:30, and
ends on Friday (20 June) at 16:00.
The full schedule is shown below.
(Tue) 17/06 INB0114 (Wed) 18/06 LMS0005
| 12:30-1:00pm |
Arrival & Registration |
|
10:30-11:30am |
Stephan Tornier, Lecture 1 |
| 1:00-1:25pm |
Greetings & Society Business |
|
11:30am-12:00pm |
Coffee break |
| 1:30-2:30pm |
Francesco Fournier-Facio |
|
12:00-1:00pm |
Anja Meyer |
| 2:30-3:15pm |
Coffee break |
|
1:00-2:30pm |
Lunch break |
| 3:15-4:15pm |
Simon Smith |
|
2:30-3:30pm |
Oihana Garaialde Ocaña, Lecture 1 |
| 4:15-5:15pm |
Oihana Garaialde Ocaña |
|
3:30-4:00pm |
Coffee break |
| 5:15-6:15pm |
Reception |
|
4:00-4:30pm |
Lukas Vandeputte |
| 6:30pm |
Society Dinner at Cozy Club Lincoln |
|
4:30-5:30pm |
Simon Smith, Lecture 1 |
(Thu) 19/06 INB1103 (Fri) 20/06 INB1103
| 10:00-11:00am |
Simon Smith, Lecture 2 |
|
09:00-10:00am |
Oihana Garaialde Ocaña, Lecture 3 |
| 11:00-11:30am |
Coffee break |
|
10:00-10:30am |
Coffee break |
| 11:30am-12:30pm |
Stacey Law |
|
10:30-11:30am |
Rudradip Biswas |
| 12:30-2:00pm |
Lunch break |
|
11:30am-12:30pm |
Simon Smith, Lecture 3 |
| 2:00-3:00pm |
Stephan Tornier, Lecture 2 |
|
12:30-2:00pm |
Lunch break |
| 3:00-3:30pm |
Coffee break |
|
2:00-2:30pm |
Sofiya Yatsyna |
| 3:30-4:00pm |
Max Gheorghiu |
|
2:30-3:00pm |
Coffee break |
| 4:00-5:00pm |
Oihana Garaialde Ocaña, Lecture 2 |
|
3:00-4:00pm |
Stephan Tornier, Lecture 3 |
Abstracts
Plenary Talks on Tuesday
Finitely generated simple groups
A central topic in group theory is the study of finitely generated simple groups. While the finite ones are classified, we are far from a general understanding of the finitely generated infinite ones, therefore this study largely consists of understanding which kinds of properties finitely generated simple groups can have, by building examples. I will do an overview of what is known and present many open questions. Time permitting, I will touch over recent joint works with Sun, with Coulon and Ho, and with Belk, Hyde and Zaremsky.
The structure of infinite permutation groups and locally compact groups
In this talk I will describe some of the rich interplay between a certain class of infinite permutation groups and totally disconnected locally compact groups, highlighting one way in which this interplay can be used to understand the structure of infinite primitive permutation groups that enjoy a natural "local finiteness" condition.
On Cohomology of Groups
In this talk we will be interested in studying problems related to cohomology of groups of different flavors.
In the first part, we will motivate the study of cohomology of a finite \(p\)-groups over the finite field of \(p\) elements. Here, as usual, \(p\) denotes a prime number. We give its definition and a list of examples to illustrate how intrincate its computation is. The last part of the talk studies two types of problems in group cohomology: clustering finite \(p\)-groups by their cohomology algebras, and studying cohomology algebra invariants such as the Krull dimension or the depth.
This talk will also serve as an introduction to some results that will be studied in the minicourse of the upcoming workshop.
Minicourses
Groups acting on trees
Groups acting on trees play an important role among general totally disconnected locally compact (t.d.l.c.) groups for theoretical and practical reasons, which we lay out over three lectures.
1. Due to the Cayley-Abels graphs construction, every compactly generated t.d.l.c. group acts vertex-transitively on a connected locally finite graph with compact open vertex stabilisers. Although explicit descriptions of Cayley-Abels graphs are hard to come by, they illustrate the prevalence of groups acting on graphs among t.d.l.c. groups. Moreover, the action lifts to the universal cover of the Cayley-Abels graph, which is a regular tree.
2. With the goal of classifying (subclasses of) groups acting on regular trees, various additional properties are typically assumed. We introduce a family of independence properties due to Tits and Banks-Elder-Willis, along with basic examples, which serve as an organising principle among general groups acting on trees. Time permitting, we also survey classification results assuming various transitivity properties.
3. We introduce a versatile class of groups acting on trees with prescribed local actions that satisfy one of the independence properties above, known as generalised Burger-Mozes groups. We prove basic properties, give examples, and characterise these groups as those locally transitive groups that satisfy one of the independence properties above and contain an edge inversion of order two.
Coclass and Cohomology
Historically, much effort has been done in classifying finite groups, and in particular, finite \(p\)-groups. Here, as usual, \(p\) denotes a prime number. The astonishing result obtained by Leedham-Green, Newmann, McKay
et al. provides a structural result for finite \(p\)-groups of given coclass. Here, a finite \(p\)-group of size \(p^n\) and nilpotency class \(c\) has coclass \(n-c\).
In this course, we introduce finite \(p\)-groups and infinite pro-\(p\) groups of finite coclass so that we can state and understand the Structure Theorem by Leedham-Green. Using this result, we will show the key steps to (partially) prove a conjecture by J. F. Carlson: for a fixed prime number \(p\) and integer number \(r\), there are only finitely many isomorphism types of mod-\(p\) cohomology algebras for all finite \(p\)-groups of coclass \(r\). As we will show, the proof is strongly based on the realization of the cohomology of finite abelian \(p\)-groups.
For those that are not familiar with cohomology, it is convenient to attend the general audience talk of the LMS Midlands Regional Meeting 2025.
Infinite permutation groups and tdlc groups
In this minicourse we will explore in detail the relationship between infinite permutation groups, locally compact groups and structural graph theory (via Dunwoody’s structure trees). The interplay between these ideas underpins many of the recent breakthroughs in the theory of totally disconnected locally compact (tdlc) groups. To illustrate this we will go through parts of the proof of a result describing tdlc groups with a compact open subgroup that is maximal. The hope is that by the end of the course you will be able to understand how all key parts of the proof work. I will highlight some significant open problems along the way.
(Long) Research Talks
Finite matrix groups: cohomology and stable elements
Sylow restriction in the representation theory of finite groups
One of the central themes in the representation theory of finite groups is to understand the relationship between the characters of a finite group \(G\) and those of its local subgroups. In particular, Sylow branching coefficients describe how an irreducible character of \(G\) decomposes upon restriction to a Sylow subgroup of \(G\). In this talk, we will discuss some of their connections to broader group-theoretic questions, before presenting new results in the case of symmetric groups in joint work with E. Giannelli.
Gorenstein analogues of a projectivity criterion over group algebras
I will talk about a
recent preprint (joint with Dimitra-Dionysia Stergiopoulou) where we formulated and answered Gorenstein projective, flat, and injective analogues of a classical projectivity question for group rings under some mild additional assumptions. Although the original question, that was proposed by Jang-Hyun Jo in 2007, was for integral group rings, in our paper, we dealt with more general commutative base rings. We made use of the developments that have happened in the field of Gorenstein homological algebra over group rings in recent years, and we managed to improve and generalize several existing results from this area along the way.
(Short) Research Talks
Conjugacy separability growth
The conjugacy separability growth function measures how easily one can separate conjugacy classes in finite quotients. This function is easily seen to relate to the residual finiteness growth (which measures how easily one can separate elements in finite quotients), or to the complexity of the conjugacy problem. In both cases the conjugacy separability function gives rise to upper bounds. However, this is a one-way correspondence. Looking at word and conjugacy problems of finite alternating groups, we can construct infinite groups where residual finiteness growth and conjugacy separability growth are controlled independently.
On a generalisation of Tate cohomology
Tate cohomology has been generalised from finite groups to all groups by several authors using different constructions. Therefore, a uniform theory is needed explaining why their approaches all lead to the same conclusions. The goal of the talk is to provide a rough sketch of such a uniform and general theory. After highlighting to which settings this theory applies, the talk concludes by stating two fundamental properties of the resulting generalisation of Tate cohomology.
Homological finiteness condition for TDLC groups
Gedrich and Gruenberg (1987) introduced invariants of a ring: SILP, the supremum of the injective lengths of the projectives, and SPLI, the supremum of the projective lengths of the injectives. Particularly, they showed that given a suitable commutative ring and group, if SPLI is finite, then SILP is also finite. Emmanouil (2010) extends the result and shows that for any group, SILP and SPLI are equal over the integers. This talk will explore the development of an analogous theory for the category of totally disconnected locally compact groups and discuss whether projective and injective modules are really dual.
