# Asymptotic Group Theory and Model Theory - a two-day workshop

Lead Research Organisation:
Royal Holloway, University of London

Department Name: Mathematics

### Abstract

The aim of this proposal is twofold. Firstly, we want to organise a two-day international workshop on "Asymptotic Group Theory and Model Theory" at Royal Holloway, University of London, in March 2012. Secondly, we want to support a six-day research visit - including the two days of the workshop - of Uri Onn (University of the Negev) and Christopher Voll (University of Bielefeld). Voll is co-organiser of the workshop and Onn one of the key speakers.

The symmetries of a mathematical, physical or chemical object - such as a graph, a molecule or a crystal - form an algebraic structure called a group. The study of finite groups led to one of the most striking achievements of 20th century mathematics: the classification of all finite simple groups. An important tool is to investigate groups by means of their linear representations, i.e. by their images as matrix groups. Asymptotic group theory, which is aimed at understanding finite and infinite groups alike, is concerned with the asymptotic properties of certain arithmetic invariants of groups. A classical direction in this comparatively young area of group theory is the study of word growth, made famous by groundbreaking work of Gromov. In another direction, the theory of representation growth, one studies infinite groups by investigating the distribution of their finite dimensional linear representations. `Zeta functions of groups' - certain infinite series which give rise to complex functions are used for encoding the arithmetic of associated growth sequences. They have proved powerful tools in developing the theory.

Over the last decades asymptotic and geometric group theory have very much benefited from the influx of techniques which come from an area of mathematical logic called model theory. Conversely, concrete problems in algebra and other classical areas, such as geometry and number theory, have stimulated important, more general model-theoretic advances. These techniques and results will form the main point of focus of the workshop, with an emphasis on three concrete topics, which can be described by the keywords: approximate groups, limit groups and motivic integration.

We envisage that the workshop will bring together researchers with quite distinct backgrounds in group theory, model theory and cognate disciplines. The meeting will allow the participants to exchange ideas and tools in rapidly expanding areas at the interface of group theory and model theory, and to learn from one another. The workshop will form part of the `South England Profinite Groups Meetings', organised by a group of mathematicians who share an interest in profinite groups. They hold about three meetings per year, dedicated to research topics of particular interest which are presented at an accessible level to younger researchers like PhD students and postdocs. The workshop is designed to be of particular benefit to younger mathematicians whose primary background is in group theory. We have deliberately chosen to invite speakers of varied research backgrounds to draw a wide picture of the relevant material.

Two participants of the meeting, Onn and Voll, will stay for four extra days to work with the investigator. The aim of their visit will be to draw immediate benefits from the workshop in ongoing joint projects on representation zeta functions of groups and the planning of further research collaborations.

The symmetries of a mathematical, physical or chemical object - such as a graph, a molecule or a crystal - form an algebraic structure called a group. The study of finite groups led to one of the most striking achievements of 20th century mathematics: the classification of all finite simple groups. An important tool is to investigate groups by means of their linear representations, i.e. by their images as matrix groups. Asymptotic group theory, which is aimed at understanding finite and infinite groups alike, is concerned with the asymptotic properties of certain arithmetic invariants of groups. A classical direction in this comparatively young area of group theory is the study of word growth, made famous by groundbreaking work of Gromov. In another direction, the theory of representation growth, one studies infinite groups by investigating the distribution of their finite dimensional linear representations. `Zeta functions of groups' - certain infinite series which give rise to complex functions are used for encoding the arithmetic of associated growth sequences. They have proved powerful tools in developing the theory.

Over the last decades asymptotic and geometric group theory have very much benefited from the influx of techniques which come from an area of mathematical logic called model theory. Conversely, concrete problems in algebra and other classical areas, such as geometry and number theory, have stimulated important, more general model-theoretic advances. These techniques and results will form the main point of focus of the workshop, with an emphasis on three concrete topics, which can be described by the keywords: approximate groups, limit groups and motivic integration.

We envisage that the workshop will bring together researchers with quite distinct backgrounds in group theory, model theory and cognate disciplines. The meeting will allow the participants to exchange ideas and tools in rapidly expanding areas at the interface of group theory and model theory, and to learn from one another. The workshop will form part of the `South England Profinite Groups Meetings', organised by a group of mathematicians who share an interest in profinite groups. They hold about three meetings per year, dedicated to research topics of particular interest which are presented at an accessible level to younger researchers like PhD students and postdocs. The workshop is designed to be of particular benefit to younger mathematicians whose primary background is in group theory. We have deliberately chosen to invite speakers of varied research backgrounds to draw a wide picture of the relevant material.

Two participants of the meeting, Onn and Voll, will stay for four extra days to work with the investigator. The aim of their visit will be to draw immediate benefits from the workshop in ongoing joint projects on representation zeta functions of groups and the planning of further research collaborations.

### Planned Impact

Since the proposed project aims to advance basic science, the immediate and measurable impact will be of an academic nature. The critical pathway towards wider economic and societal impact is primarily given

(i) through the training of young scientists, who will eventually contribute to the UK economy through careers inside and outside academia,

(ii) raising the profile of the experienced scientists involved and of the host institution, thereby strengthening the economic prospects of the latter,

(iii) the creation of a new international network between researchers in Belgium, France, Germany and the UK, with a view towards future grant applications (e.g. under the ERC Synergy Grant scheme),

(iv) through long-term effects, which are difficult to predict, but often stimulated by the exchange of ideas with cognate disciplines, such as computer science.

The process referred to in (iv) can best be illustrated by giving a concrete example. One of the topics addressed in the workshop, viz. expansion in non-abelian finite simple groups, is related to the construction of expander graphs. This forms a bridge to more applied research on network design, thus leading to practical applications. Loosely speaking, the basic underlying problem is to construct a highly connected network without using up too many resources. Expander constructions have applications, for instance, in the design of robust computer networks and the construction of error-correcting codes. The latter are used for transmitting data over an unreliable communication channel without loss of fidelity. It is surprising and beautiful that the abstract study of non-abelian finite simple groups leads to families of expanding graphs.

We aim to generate and maximise the academic and non-academic impact of the workshop by

- bringing together experts from several international institutions and from different areas of mathematics,

- disseminating workshop material, such as summaries of talks and slide presentations, using a dedicated workshop web site.

(i) through the training of young scientists, who will eventually contribute to the UK economy through careers inside and outside academia,

(ii) raising the profile of the experienced scientists involved and of the host institution, thereby strengthening the economic prospects of the latter,

(iii) the creation of a new international network between researchers in Belgium, France, Germany and the UK, with a view towards future grant applications (e.g. under the ERC Synergy Grant scheme),

(iv) through long-term effects, which are difficult to predict, but often stimulated by the exchange of ideas with cognate disciplines, such as computer science.

The process referred to in (iv) can best be illustrated by giving a concrete example. One of the topics addressed in the workshop, viz. expansion in non-abelian finite simple groups, is related to the construction of expander graphs. This forms a bridge to more applied research on network design, thus leading to practical applications. Loosely speaking, the basic underlying problem is to construct a highly connected network without using up too many resources. Expander constructions have applications, for instance, in the design of robust computer networks and the construction of error-correcting codes. The latter are used for transmitting data over an unreliable communication channel without loss of fidelity. It is surprising and beautiful that the abstract study of non-abelian finite simple groups leads to families of expanding graphs.

We aim to generate and maximise the academic and non-academic impact of the workshop by

- bringing together experts from several international institutions and from different areas of mathematics,

- disseminating workshop material, such as summaries of talks and slide presentations, using a dedicated workshop web site.

## People |
## ORCID iD |

Benjamin Klopsch (Principal Investigator) |

### Publications

*Representation zeta functions of compact $p$ -adic analytic groups and arithmetic groups*in Duke Mathematical Journal