**LMS Midlands Regional Meeting and Workshop**

2nd – 4th June 2021, University of Lincoln

LMS Midlands Regional Meeting

2nd June 2021, University of Lincoln

The regional meeting will consists of three talks, by Professor Delaram Kahrobaei (York), by Dr Simon Smith (Lincoln), and by Professor Alexandre Borovik (Manchester). The last talk will be of a public lecture style, and this talk will be followed by a virtual drinks reception and a poster session.

All talks for the regional meeting will take place online via Microsoft Teams.

**Regional meeting programme:**

**Wednesday, 2nd June 2021**

**13.30 – 13.45:** LMS Welcome

**13.45 – 14.45:** Delaram Kahrobaei (University of York) – *Title: Cryptographic Multilinear Map via Pro-p Groups*

**14.45 – 15.15:** break

**15.15 – 16.15:** Simon Smith (University of Lincoln) – *Title: A local-to-global complement to Bass-Serre Theory*

**16.30 – 17.30:** Alexandre Borovik (University of Manchester) – *Title: Black Box Algebra and Homomorphic Encryption*

**17.40 – 18.10:** Virtual drinks reception and poster session

The abstracts for the talks may be found further below.

To register for the regional meeting and to receive the talk links, please see the contact details below. Please also indicate if you would like to present a poster.

Workshop: Profinite Groups and Related Aspects

3rd – 4th June 2021, University of Lincoln

Profinite groups are central objects in group theory, equipped with an interesting topological structure. Enlarging the picture, one sees profinite groups as a subclass of totally disconnected locally compact groups. On the other hand, a key subclass of profinite groups, the pro-*p* groups, have important connections in the theory of finite *p*-groups, in Lie theory, especially via the *p*-adic analytic pro-*p* groups. This workshop brings together all aspects of, and relating to, profinite groups, and promotes new connections between the aspects.

All talks for the workshop will take place online via Microsoft Teams, and the workshop will begin on the morning of Thursday the 3rd of June 2021, and end at 12.30pm on Friday the 4th of June 2021.

To register for the workshop and to receive the talk links, please see the contact details below

**Speakers:**

Gunnar Traustason (University of Bath)

Anastasia Hadjievangelou (University of Bath)

Nadia Mazza (Lancaster University)

John Wilson (University of Cambridge and University of Leipzig)

Pavel Shumyatsky (University of Brasilia)

Colin Reid (University of Newcastle, Australia)

Alejandra Garrido (Autonomous University of Madrid)

Henry Bradford (University of Cambridge)

Rachel Camina (University of Cambridge)

**Workshop programme:**

**Thursday, 3rd June 2021**

**9.00 – 9.50:** Gunnar Traustason – *Title*: *Powerfully nilpotent groups*

**10.00 – 10.50:** John Wilson – *Title*: *A first-order perspective on finite groups*

**10.50 – 12.00:** break

**12.00 – 12.50:** Nadia Mazza – *Title*: *Mackey functors for profinite groups and beyond *

**12.50 – 14.00:** break

**14.00 – 14.50:** Henry Bradford – *Title: Quantitative LEF and topological full groups*

**14.50 – 16.00:** break

**16.00 – 16.50:** Rachel Camina – *Title: Normal cyclic coverings of finite p-groups *

**17.00 – 17.50:** Pavel Shumyatsky – *Title: On profinite groups with restricted centralizers of commutators *

**Friday, 4th June 2021**

**9.00 – 9.50:** Alejandra Garrido – *Title*: *New examples of simple locally compact groups *

**10.00 – 10.50:** Anastasia Hadjievangelou – *Title*: *Left 3-Engel Elements in Locally Finite p-Groups*

**10.50 – 12.00:** break

**12.00 – 12.50:** Colin Reid – *Title*: *Embedding totally disconnected locally compact groups into simple groups*

Registration and Contact

For queries and to register (for the regional meeting and/or the workshop) please email Anitha Thillaisundaram (athillaisundaram at lincoln dot ac dot uk).

Talk Abstracts

**For the regional meeting:**

**Delaram Kahrobaei** – *Title: Cryptographic Multilinear Map via Pro-p Groups*

Abstract: In recent years, algorithmic problems in group theory have been proposed for use in cryptographic protocols that are promising in post-quantum cryptosystems. Various groups and problems have been considered. Among them, *p*-groups are one of the promising candidates. In this talk, I survey some of these schemes. In particular, I discuss a joint work with Mima Stanojkovski (Max Planck Institute) on Cryptographic multilinear maps using pro-*p* groups. To any nilpotent group of class *n*, one can associate a non-interactive key exchange protocol between *n*+1 users. The multilinear commutator maps associated to nilpotent groups play a key role in this protocol. In the present paper, we analyze the security of this key exchange when applied to finite *p*-groups, both in the generic case and for explicit families of groups. We show, moreover, how infinite pro-p groups can be employed as platforms for any number of users.

**Simon Smith** – *Title: A local-to-global complement to Bass-Serre Theory*

Abstract: Groups acting on trees play a fundamental role in the theory of groups. Bass-Serre Theory, and in particular the notion of a graph of groups, is a powerful tool for decomposing groups acting on trees. It is also useful for constructing discrete groups acting on trees. However, its usefulness for constructing non-discrete groups acting on trees is, in some situations, severely limited. Such groups play an important role the theory of locally compact groups, as they are a rich source of examples of compactly generated simple groups.

An alternative, but complementary, approach to the study of groups acting on trees has recently emerged based on local actions. Its origins can be traced back to the work of J. Tits, but it was in a seminal paper of M. Burger and Sh. Mozes that this “local-to-global” approach was first fully articulated. Typically, these local-to-global constructions have something called Tits’ independence property (P). Intuitively this property means that a group can act independently on different branches of the tree.

In joint work with Colin Reid, we have developed a general method for describing and classifying all actions of groups on trees with property (P). This is done using an object called a local action diagram, akin to a graph of groups, but for local actions. Our work can be seen as a `local action’ complement to Bass-Serre theory. As an example of how effective this local action diagram approach is, for a group *G* with property (P) one can easily determine whether or not *G* has compact generation and simplicity directly from its local action diagram.

**Alexandre Borovik** – *Title: Black Box Algebra and Homomorphic Encryption*

Abstract: (Joint work with Sukru Yalcinkaya) We offer a systematic approach to a class of attacks on communication

channels protected by homomorphic encryption. These attacks are based on black box algebraic analysis. Our conclusion is that wide classes of algebraic structures should not be used as ambient structures for homomorphic encryption. We give some examples for groups and rings, but our general methodology is much wider applicable.

Black box algebra deals with a category where objects are finite algebraic structures (fields, rings, group,s projective planes etc.)

with elements implemented as 0-1 strings of length L (perhaps different for different objects) and operations are performed by external devices or algorithms which work in time bounded by a polynomial in L). Similarly, morphisms are homomorphisms computable in polynomial time. We will show that this is a fascinating theory with many unusual features and a huge range of open problems.

**For the workshop:**

**Gunnar Traustason** – *Title*: *Powerfully nilpotent groups *

Abstract: In this talk we discuss a special subclass of powerful groups called powerfully nilpotent groups. These are finite *p*-groups that possess a central series of a special kind. We will describe some structure theory and a ‘classification’ in terms of an ancestry tree and powerful coclass. We will also talk about some other related subclasses of powerful groups.

**John Wilson** – *Title*: *A first-order perspective on finite groups*

Abstract: The finite axiomatizability of classes of finite groups, and the definability of naturally occurring subgroups, have attracted considerable attention. In this talk, some of the results, positive and definite, will be discussed, and it will be shown that the strikingly different behaviour of certain properties seems to be reflected in (non-first-order) studies of these properties.

**Nadia Mazza** – *Title*: *Mackey functors for profinite groups and beyond *

Abstract: Mackey functors have been studied in-depth for finite groups, providing a useful framework to study various aspects of group theory, such as group cohomology for instance. In this talk, we will revisit the subject, presenting some progress towards a theory of Mackey functors for profinite groups, and more generally tdlc groups. This is work in progress with Ilaria Castellano and Brita Nucinkis.

**Henry Bradford** – *Title: Quantitative LEF and topological full groups*

Abstract: Topological full groups of minimal subshifts are an important source of exotic examples in geometric group theory, as well as being powerful invariants of symbolic dynamical systems. In 2011, Grigorchuk and Medynets proved that TFGs are LEF, that is, every finite subset of the multiplication table occurs in the multiplication table of some finite group. In this talk we explore how the growth of the finite groups which occur reflects asymptotic properties of the associated subshift. Joint work with Daniele Dona.

**Rachel Camina** – *Title: Normal cyclic coverings of finite p-groups*

Abstract: A covering of a group $G$ is a set of proper subgroups $\{ H_i \}$, called components, such that $G = \bigcup H_i$. Identifying coverings of groups and the minimal size of such a covering has a long mathematical history. More recently authors have considered normal coverings, that is coverings that are invariant under $G$-conjugation. The normal covering number is the smallest number of conjugacy classes of proper subgroups in a normal covering of $G$. Along with Mariagrazia Bianchi, Emanuele Pacifici and Mark Lewis I have been considering the normal covering number of finite $p$-groups where all components are cyclic. The work was motivated by a

question in Timm von Puttkamer’s PhD thesis which considered the finiteness of the classifying spaces for virtually cyclic subgroups.

**Pavel Shumyatsky** – *Title: On profinite groups with restricted centralizers of commutators *

Abstract: A group *G *has restricted centralizers if for each *g *in *G* the centralizer C_*G*(*g*) either is finite or has finite index in *G.* A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In this talk I will describe the theorem that if *w* is a multilinear commutator word and *G *a profinite group with restricted centralizers of *w*-values, then the verbal subgroup *w*(*G*) is abelian-by-finite.

**Alejandra Garrido** – *Title*: *New examples of simple locally compact groups *

Abstract: Profinite groups can be studied from the point of view of their commensurators — groups that induce an isomorphism between open subgroup. Such commensurators then inherit the topology of the profinite group and become totally disconnected locally compact groups. Examples include the $p$-adic integers inside the $p$-adic rationals, the group of automorphisms of an infinite rooted regular tree inside the group of almost automorphisms of the tree.

The general theory of totally disconnected locally compact groups has advanced by leaps and bounds in recent years. Yet there is a pressing need for examples of such groups that are simple and compactly generated. One such is the group of almost automorphisms of a rooted regular tree. This turns out to also be an example of a piecewise full group (a.k.a. topological full group) of homeomorphisms of the Cantor set (the boundary of the tree). These piecewise full groups have been a source of new examples of finitely generated infinite simple groups.

I will report on joint work with Colin Reid and David Robertson on when and how piecewise full groups yield new examples of compactly generated, simple, totally disconnected locally compact groups.

**Anastasia Hadjievangelou** – *Title*: *Left 3-Engel Elements in Locally Finite p-Groups*

Abstract: Engel Theory is of significant interest in group theory as there is an unmistakable correlation between Engel and Burnside problems. In this talk we first introduce Engel elements and Engel groups and in particular we expand our knowledge on locally finite Engel groups. It is important to know that M. Newell proved that if x is a right 3-Engel element in a group G then a lies in HP(G) (Hirsch-Plotkin radical) and in fact he proved the stronger result that the normal closure of x is nilpotent of class at most 3. The natural question arises whether the analogous result holds for left 3-Engel elements. We will give various examples of locally finite p-groups G containing a left 3-Engel element x whose normal closure is NOT nilpotent. (This is joint work with Gunnar Traustason and Marialaura Noce.)

**Colin Reid** – *Title*: *Embedding totally disconnected locally compact groups into simple groups*

Abstract: Let S be the class of totally disconnected locally compact (tdlc) groups that are compactly generated and nondiscrete. The class S can be compared on the one hand with the class of simple Lie groups, where the complete list of isomorphism classes is known, and on the other with the class of finitely generated simple discrete groups, which is known to be “wild”; for example, it was shown in the 1970s that every countable group is a subgroup of a 2-generator simple group. By contrast, we do not really know how “large” S is: for instance, we know (due to Smith) it has uncountably many isomorphism classes, but it is unknown whether it has uncountably many local isomorphism classes. Analogous to the situation for countable groups, one might ask if every tdlc group is a closed subgroup, perhaps even an open subgroup, of a group in S. As stated the answer is “no” due to several known obstacles. However, what I will present is a general construction for “almost” embedding compactly generated tdlc groups, in groups that are “almost” in S, as “almost” open subgroups. The construction is part of an ongoing project with Garrido, Robertson and myself on locally compact piecewise full groups, and also uses Smith’s construction.

Regional Meeting and Workshop Participants

Alan Camina (University of East Anglia)

Alejandra Garrido (Autonomous University of Madrid)

Alexandre Borovik (University of Manchester)

Alireza Abdollahi (University of Isfahan)

Anastasia Hadjievangelou (University of Bath)

Andoni Zozaya (University of the Basque Country)

Avinoam Mann (University of Jerusalem)

Ayse Berkman (Mimar Sinan Guzel Sanatlar University)

Brita Nucinkis (Royal Holloway, London)

Cristina Acciarri (University of Brasilia)

Colin Reid (University of Newcastle, Australia)

Dan Segal (University of Oxford)

David Bradley-Williams (Heinrich Heine University of Duesseldorf

Delaram Kahrobaei (University of York)

Elena Di Domenica (University of the Basque Country and University of Trento)

Emilio Pierro (London School of Economics)

Gulin Ercan (Middle East Technical University)

Gunnar Traustason (University of Bath)

Henry Bradford (University of Cambridge)

Ilaria Castellano (University of Milan-Bicocca)

Inna Capdeboscq (University of Warwick)

Jan-Christoph Schlage-Puchta (University of Rostock)

John Wilson (University of Cambridge and University of Leipzig)

Jone Uria Albizuri (University of the Basque Country)

Karthika Rajeev (Heinrich Heine University of Duesseldorf)

Luca Maria Di Gravina (University of Milan-Bicocca)

Marialaura Noce (Georg August University Goettingen)

Marta Morigi (University of Bologna)

Martina Conte (Heinrich Heine University of Duesseldorf)

Martyn Quick (University of St Andrews)

Mike Ogiugo (Yaba College of Technology, Lagos)

Nadia Mazza (Lancaster University)

Oihana Garaialde (University of the Basque Country)

Patrizia Longobardi (University of Salerno)

Paula Macedo Lins de Araujo (KU Leuven)

Pavel Shumyatsky (University of Brasilia)

Rachel Camina (University of Cambridge)

Roman Gorazd (University of Newcastle, Australia)

Shrinit Singh (Shiv Nadar University)

Simone Blumer (University of Milan-Bicocca)

Thomas Smith (University of Lincoln)

Yiftach Barnea (Royal Holloway, London)

Acknowledgements

This meeting is supported by the LMS Regional Meeting Grant, and by the School of Mathematics and Physics of the University of Lincoln.