Introductions to Groups St Andrews Proceedings

The Groups St Andrews Conferences have been held every four years beginning in 1981. The Proceedings of each of the conferences were published by Cambridge University Press. You can see the Introductions to these proceedings below.

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1981 1985 1989 1993 1997
2001 2005 2009 2013 2017

GROUPS ST ANDREWS 1981

INTRODUCTION

An international conference 'Groups - St Andrews 1981' was held in the Mathematical Institute, University of St Andrews during the period 25th July to 8th August 1981. The main topics of the conference: combinatorial group theory; infinite groups; general groups, finite or infinite; computational group theory are all well-represented in the survey and research articles that form these Proceedings. Four courses each providing a five-lecture survey, given by Joachim Neubüser, Derek Robinson, Sean Tobin and Jim Wiegold have been expanded, subsequently, into articles forming the first four chapters of the volume. Many of the themes in these chapters recur in the survey and research articles which form the second part of the volume.

Methods and techniques such as homology, geometrical methods and computer implementation of algorithms are used to obtain group theoretical results. Computational methods are surveyed in several articles in particular the major survey by Joachim Neubüser and find application in papers on Burnside groups and finite simple groups. In fact Burnside groups are discussed in two rather different papers, a survey of groups of exponent four by Sean Tobin and a major contribution to the exponent five case by Marshall Hall and Charles Sims. Derek Robinson exploits the way in which cohomology groups arise in group theory to establish some splitting and near-splitting theorems. Rudolf Beyl also uses homological techniques to discuss group extensions. The Schur multiplicator which arises naturally in this context is given a 'nonhomological' treatment (and is called the Schur multiplier!) in a survey by Jim Wiegold.

Splitting results are also studied by Klaus Roggenkamp when he considers the splitting of the natural injection from a group to the group of units in its group ring. The structure and group-theoretical properties of the group of units of a group ring feature in another survey article. Presentations of groups are studied in many of the articles already described. Several other authors discuss groups with presentations of a specific type, for example small cancellation groups and one-relator groups.

The reader will find extensive bibliographies with many of the papers. Many open problems are also cited in the papers often with possible methods of attack. We hope that this not only makes the volume a useful record of the current state of the art but also points the way to future developments.

During the two weeks of the conference, group theory programmes on the Aberdeen Honeywell and the St Andrews VAX were widely used by conference participants. The CAYLEY group theory package was demonstrated from the Queen Mary College computer via a link to the St Andrews VAX. We would like to thank those in the University of St Andrews Computing Laboratory who helped make these facilities available.

We would like to express our thanks for the assistance we received from our colleagues in the Mathematical Institute and in particular John Howie, John O'Connor and Peter Williams. We thank the British Council for grants for two conference participants and the London Mathematical Society for their early pledge of financial support which enabled the conference to proceed. Our thanks are also given to the London Mathematical Society and Cambridge University Press for their help and encouragement in the preparation of this volume, and to Shiela Wilson for so willingly undertaking the daunting task of typing a volume of this length and for the high quality of the final typescript.

Our final thanks go to those who have contributed articles to this volume. We have edited these articles to produce some uniformity without, we hope, destroying individual styles. We will have introduced inevitably errors into the text. For these errors we take full responsibility.

Colin M. Campbell
Edmund F. Robertson
St Andrews, June 1982

GROUPS ST ANDREWS 1985

INTRODUCTION

An international conference 'Groups - St Andrews 1985' was held in the Mathematical Institute, University of St Andrews, Scotland during the period 27 July to 10 August 1985. The initial planning for the conference began in August 1981 and in May 1983 invitations were given to Professor Seymour Bachmuth (Santa Barbara, California), Professor Gilbert Baumslag (City University, New York), Dr Peter Neumann (Queen's College, Oxford), Dr James Roseblade (Jesus College, Cambridge) and Professor Jacques Tits (Collège de France, Paris). They all accepted the invitation to give lecture courses consisting of four or five lectures each and these courses formed the main part of the first week of the conference. They have all contributed major articles to this volume based on these lecture courses. Automorphisms of soluble groups, the topic of Seymour Bachmuth's lectures, is written up in two articles, the second being contributed by his co-worker Horace Mochizuki.

In the second week of the conference there were twelve one-hour invited lectures. This volume contains articles based on eight of these talks. In addition there was a full programme of seminars, the remaining articles of this book arising from these lectures. A companion volume to this book, published as a part of the Proceedings of the Edinburgh Mathematical Society, contains articles based on two of the one-hour invited lectures and a number of articles based on some of the research seminars.

One of the major features of this book is the number of surveys, written by leading researchers, in the wide range of group theory covered. Jim Roseblade gives a group theorist's view of group rings in his survey describing work on group rings which has been motivated by problems in group theory. Jacques Tits provides a survey on buildings and group amalgamations considerably more major than his modest description of it as 'but a brief summary of five lectures ...' would suggest.

References form an important component of any survey and since the majority of articles in this book are surveys there is a wealth of references. Gilbert Baumslag's article on one-relator groups wins the prize, containing over one hundred references. Only by taking the union of the references of two surveys on quite different aspects of the modular group does one get a more extensive bibliography.

The computing aspect of group theory is evident in a number of papers in this volume, in particular in the paper by Peter Neumann describing some algorithms for computing with finite permutation groups. At the conference this computing aspect was served by the availability of the CAYLEY group theory package and other group theory programs resident on the St Andrews VAX. These facilities were well used and we would like to thank the Computing Laboratory of the University of St Andrews for their assistance.

Groups - St Andrews 1985 received financial support from the Edinburgh Mathematical Society, the London Mathematical Society and the British Council. We gratefully acknowledge this financial support and would like to thank the Edinburgh Mathematical Society and the London Mathematical society for their help with publishing. A total of 366 people from 43 different countries registered for the conference.

We would like to express our thanks to all our colleagues who helped in the running of the conference and in particular Tom Blyth, Peter Sumner and Trevor Walker. We would also like to thank our wives for their help and forbearance. We would like to thank Shiela Wilson who acted as conference secretary for the two weeks of the conference and who has undertaken the major task of transforming almost 800 pages of A4 typescript into the 350 camera-ready pages for this volume.

Our final thanks go to the authors who have contributed articles both to this volume and to the companion part in the Proceedings of the Edinburgh Mathematical Society. Because of space restrictions and our desire to include as many articles as possible, we have in some cases reduced the number of displays in our attempt to save space. We apologise to the authors and trust that this will not detract from the articles.

Edmund F. Robertson
Colin M. Campbell

GROUPS ST ANDREWS 1989

INTRODUCTION

An international conference 'Groups - St Andrews 1989' was held in the Mathematical Institute, University of St Andrews, Scotland during the period 29 July to 12 August 1989. A total of 293 people from 37 different countries registered for the conference. The initial planning for the conference began in July 1986 and in the summer of 1987 invitations were given to Professor J A Green (Warwick), Professor N D Gupta (Manitoba), Professor O H Kegel (Freiburg), Professor A Yu Ol'shanskii (Moscow) and Professor J G Thompson (Cambridge). They all accepted our invitation and gave courses at the conference of three or four lectures. We were particularly pleased that Professor Ol'shanskii was able to make his first visit to the West. The above courses formed the main part of the first week of the conference. All the above speakers have contributed articles based on these courses to the Proceedings.

In the second week of the conference there were fourteen one-hour invited survey lectures and a CAYLEY workshop with four main lectures. In addition there was a full programme of research seminars. The remaining articles in the two parts of the Proceedings arise from these invited lectures and research seminars.

The two volumes of the Proceedings of Groups - St Andrews 1989 are similar in style to 'Groups - St Andrews 1981' and 'Proceedings of Groups - St Andrews 1985' both published by Cambridge University Press in the London Mathematical Society Lecture Notes Series. Rather than attempt to divide the two parts by an inevitably imprecise division by subject area we have divided the two parts by author name, the first part with author names beginning A-G and the second part author names beginning H-V.

A feature of these volumes is the number of surveys written by leading researchers, in the wide range of group theory covered. From the papers an extensive list of references may be built up and some of the diversity of group theory appreciated.

The computing aspect of group theory was catered for in several ways. As mentioned above there was a series of lectures on CAYLEY and a CAYLEY workshop; there was also a lecture on GAP. Participants were also given access to the Microlaboratory equipped with MAC's, to the University of St Andrews VAX's and to a SUN 3/260 running a wide range of software including CAYLEY, GAP, SPAS and SOGOS. All these facilities were well used by the conference participants.

Groups - St Andrews 1989 received financial support from the Edinburgh Mathematical Society, the London Mathematical Society and the British Council. We gratefully acknowledge this financial support. We would also like to thank Cambridge University Press and the London Mathematical Society for their help with publishing.

We would like to express our thanks to all our colleagues who helped in the running of the conference and in particular Patricia Heggie, John O'Connor and Trevor Walker. We would also like to thank our wives for their help and forbearance. We would like to thank Shiela Wilson for so willingly undertaking the daunting task of typing the two volumes of the Proceedings, having already typed the Proceedings of the previous two St Andrews conferences.

Our final thanks go to those authors who have contributed articles to these volumes. We have edited these articles to produce some uniformity without, we hope, destroying individual styles. For any inconsistency in, and errors introduced by, our editing we take full responsibility.

Colin M Campbell
Edmund F Robertson

GROUPS ST ANDREWS 1993

INTRODUCTION

An international conference 'Groups 1993 Galway / St Andrews' was held at University College, Galway, Ireland during the period 1 to 14 August 1993. This followed in the main the successful format developed in 1981, 1985 and 1989 by C M Campbell and E F Robertson. They invited T Hurley, S Tobin and J Ward to join them and continue the series in 1993 in Galway. Serious planning got under way when the five organisers met at a Warwick conference in March 1991, and decided to invite as principal speakers Professor J L Alperin (Chicago), Professor M Broué (Paris), Professor P H Kropholler (London), Professor A Lubotzky (Jerusalem) and Professor E I Zelmanov (Madison). All of these agreed to give courses of about five lectures each, and articles based on these courses form a valuable part of these Proceedings - particularly so as in some cases they are strongly complementary in subject matter. Also, as it transpired, one speaker was awarded a Fields Medal exactly one year later at the 1994 ICM in Zurich; the organisers have great pleasure in congratulating Professor Zelmanov most heartily - and hope perhaps that this may augur well for future speakers in the series!

An invitation to Professor J Neubüser (Aachen) to arrange a workshop on Computational Group Theory and the use of GAP was taken up so enthusiastically by him that the workshop became effectively a fully-fledged parallel meeting throughout the second week, with over thirty hours of lectures by experts and with practical sessions organised by M Schönert (Aachen). These Proceedings contain an article by Professor Neubüser based on a lecture he gave in the first week of the conference.

In addition there were sixteen invited one-hour lectures, and a very great number of research talks, which reflected the fact that the meeting was attended by 285 mathematicians (who were accompanied by about 150 family members) from 35 countries around the world. The articles in these Proceedings, other than those of the main speakers, are based on one-hour invited lectures and other research talks.

As in 1989 there are two volumes of the Proceedings, with the contributions arranged according to author's names in alphabetical order from A in Volume One to Z in Volume Two, except for a minor perturbation in alphabetical order at the join between the two volumes. They will form a stimulating record of recent work as well as being a valuable reference source, due to the wealth of material covered in the main courses. We thank all our contributors, and regret that we could not - because of space restrictions - accept several other worthy papers.

An unusual feature of this conference was the setting aside of one day for a special programme of lectures to honour the 65th birthday of Professor K W Gruenberg, in recognition of his many contributions to group theory. A second feature was the publication by the Galway Mathematics Department of a splendid memoir by Professor Lubotzky on Subgroup Growth, prepared by him as a background for his course of lectures. Another feature was the videoing of all the talks of the main speakers which form a lovely record of the events. These videos may still be obtained from the Galway Mathematics Department.

There are many who helped to make the conference a memorable occasion; in particular we thank the staff of the Computing Services in University College Galway, our colleagues and students, members of the Secretariat in both Galway and St Andrews, and members of our own families who cheerfully helped out in various ways. We extend our thanks to the Administration of University College Galway for the Reception in the Aula Maxima; and we are grateful to the Mayor of Galway who gave a special reception in the Corporation's Council Chamber.

We gratefully acknowledge financial support for the academic program from the London Mathematical Society, the Royal Irish Academy, the Irish Mathematical Society, An Bord Gais, Irish Shell, Allied Irish Banks, Bank of Ireland, Bord Failte, the International Science Foundation, University College Galway, the University of St Andrews and the Deutsche Forschungsgemeinschaft.

As our final word, an focal scuir, we wish to thank Cambridge University Press for help with these Proceedings and Nik Ruskuc for undertaking so willingly the enormous task of reformatting files in many flavours of TeX. As in previous volumes, the present editors have endeavoured to produce a measure of uniformity - hopefully without distorting individual styles. For any inconsistencies in, or errors introduced by, our editing we of course accept responsibility.

Colin M Campbell, Thaddeus C Hurley, Edmund F Robertson, Sean J Tobin, James J Ward

GROUPS ST ANDREWS 1997

INTRODUCTION

An international conference Groups St Andrews 1997 in Bath was held on the campus of the University of Bath during the period 26 July - 9 August 1997. Some 299 mathematicians from 41 countries were involved in the meeting, as well as 82 family members and partners. This was the fifth meeting of the four-yearly Groups St Andrews Conferences, and the series continues to flourish. The shape of the conference was similar to the previous conferences in that the first week was dominated by five series of talks, each surveying an area of rapid contemporary development in group theory. The main speakers were Laszlo Babai (Chicago), Martin Bridson (Oxford), Chris Brookes (Cambridge), Cheryl Praeger (Western Australia) and Aner Shalev (Jerusalem). The second week featured two special days, a Burnside Day and a Lyndon Day. Our thanks are due to Efim Zelmanov (Yale) and Chuck Miller III (Melbourne), respectively, for helping organise the programmes for these days. In addition the week contained a wide variety of research talks. In the evenings throughout the conference, and during the rest periods, there was an extensive social programme, only some of which was disrupted by rain. There was also much extemporised music-making in the Senior Common Room in the evenings.

These Proceedings contain the written evidence of the academic achievements of the conference. The five main speakers have all provided substantial survey articles, giving a wide perspective on their fields. In the case of Laszlo Babai, the article is written jointly with Bob Beals (Arizona), one of the invited speakers of the second week. Sixteen other papers in these Proceedings are written by authors who gave one hour invited lectures. A rigorous journal-style refereeing process was applied to all the research articles and survey articles submitted for publication in the Proceedings.

Less easy to quantify, but much more important, were the exchanges of ideas and joint work that was done, both at the conference and as a result of meetings at the conference. As the largest regular meeting on group theory in the world, this series has provided a continuing stimulus to research in group theory.

There are many who helped to make the conference a memorable occasion; in particular we thank Mrs Nada Harvey for secretarial assistance, and Mr John McDermott, Mr Aaron Wilson and many other students from the Department of Mathematical Sciences of the University of Bath whose help in the day to day running of the conference was invaluable. Many of Geoff Smith's colleagues from the academic staff of the Department of Mathematical Sciences gave valuable help and we particularly thank Professor John Toland for guidance on conference organization, and Dr Fran Burstall for TeX and LaTeX assistance to the conference and its newspaper, the Daily Group Theorist.

We record our thanks also to the then Head of School, Professor Alastair Spence and to the Pro-Vice Chancellor Professor I M Jamieson, for smoothing the path of the conference organization. For her constructive attitude and industrious work with the conference accommodation we thank Marian Short from the conference office of the University of Bath.

It is a pleasure to acknowledge the excellent level of support we received from various funding bodies. The Edinburgh Mathematical Society and the London Mathematical Society provided financial help which defrayed the main speakers' travel, registration and accommodation expenses. The London Mathematical Society additionally sponsored three visitors from Moscow State University, Professor A. Yu. Ol'shanskii, and two research students Ivan Arzhantsev and Gulnara Arzhantseva. This funding provided support for travel, accommodation, conference fees and subsistence. The Royal Society of London treated the conference as two meetings, because of its exceptional size, and because it is held only once in four years. They therefore generously supported four visitors from the former Soviet Union.

We also gratefully acknowledge the financial support from the University of Bath Initiative Fund, the Department of Mathematics of the University of Bath and the School of Mathematical and Computational Sciences of the University of St Andrews. We thank the Bath and North-East Somerset Council for hosting a civic reception at the Roman Baths. Finally, it is a pleasure to thank Olga Tabachnikova both for her assistance in countless aspects of conference organization, and for the loan of Geoff Smith.

Colin Campbell, Ed Robertson, Nik Ruskuc , Geoff Smith

GROUPS ST ANDREWS 2001

INTRODUCTION

Groups St Andrews 2001 in Oxford was another highly successful conference in the continuing series. The conference, held from 5 August 2001 to 18 August 2001, was attended by 230 mathematicians from 35 countries. The lectures and talks were given in the Mathematical Institute of the University of Oxford. We thank Lady Margaret Hall, St Anne's College, the Queen's College and Merton College for providing accommodation. We also acknowledge with gratitude the financial support of the Edinburgh and London Mathematical Societies.

The organizing committee consisted of C M Campbell (University of St Andrews), D P Groves (Merton), R P Martineau (Wadham), P M Neumann (Queens), E F Robertson (University of St Andrews), G C Smith (University of Bath), W B Stewart (Exeter) and G A Stoy (LMH). Administrative support was provided by Jan Campbell and Maureen White.

The main speakers, who were invited to give a series of talks, were Marston D E Conder (Auckland), Persi Diaconis (Stanford), Peter P Pálfy (Eötvös Lorand, Budapest), Marcus du Sautoy (Cambridge), and Michael R Vaughan-Lee (Oxford). As has become the tradition, all these invited speakers have written substantial articles for these Proceedings. All papers have been subjected to a formal refereeing process comparable to that of a major international journal. Publishing constraints have forced the editors to exclude some very worthwhile papers, and this is of course a matter of regret.

The Theory of Groups continues to move forward on many fronts, and twenty years on from the announcement of the classification of the finite simple groups, it prospers perhaps surprisingly well (rather like Mark Twain). It is a measure of the success of this conference series and this subject that mathematical libraries around the world are collecting the series of St Andrews Conference Proceedings.

As LaTeX has become the almost universal typesetting language for mathematics, the nuts and bolts of the editorial job are perhaps getting a little easier. We hope that in future even more contributors will stick faithfully to the typesetting standard prescribed by the editors (so that these hacks will have more time to do research in group theory, lie on beaches and so on). It is hoped that the next conference in this series will be held in 2005 and will be Groups St Andrews in St Andrews (revisiting the scene of the original crime), and it is also hoped that in 2009 we will return once more to Bath.

As these proceedings go to press, news has just arrived that one of the grandfathers of the subject, B H Neumann, has died. He attended the first Groups St Andrews conference in 1981 and some later ones in the series. Many of our community knew him with great affection -- his warm smile and shoestring tie will linger in the memory almost as long as his important contributions to mathematics.

CMC, ERF, GCS

GROUPS ST ANDREWS 2005

INTRODUCTION

Groups St Andrews 2005 was held in the University of St Andrews from 30 July to 6 August 2005. This was the seventh in the of Groups St Andrews group theory conferences organised by Colin Campbell and Edmund Robertson of the University of St Andrews. The first three were held in St Andrews and subsequent conferences held in Galway, Bath and Oxford before returning to St Andrews in 2005. We are pleased to say that the conference was, we believe, a success having been attended by 230 participants from 37 countries. The lectures and talks were given in the Mathematical Institute and the School of Physics and Astronomy of the University of St Andrews. Accommodation was provided in New Hall and in Fife Park.

The Scientific Organising Committee of Groups St Andrews 2005 was: Colin Campbell (St Andrews), Nick Gilbert (Heriot Watt), Steve Linton (St Andrews), John O'Connor (St Andrews), Edmund Robertson (St Andrews), Nik Ruskuc (St Andrews), Geoff Smith (Bath). The Committee received very valuable support from our Algebra colleagues at St Andrews, both staff and postgraduate students. Once again, we believe that the support of the two main British mathematics societies, the Edinburgh Mathematical Society and the London Mathematical Society has been an important factor in the success of these conferences.

The main speakers at the meeting were Peter J Cameron (Queen Mary, London), Rostislav I Grigorchuk (Texas A&M), John C Meakin (Nebraska Lincoln) and Akos Seress (Ohio State). Additionally there were seven one hour invited speakers together with an extensive programme of over a hundred seminars; a lot to fit into a week! As has become the tradition, all the main speakers have written substantial articles for these Proceedings. Each volume begins with two such articles. All papers have been subjected to a formal refereeing process comparable to that of a major international journal. Publishing constraints have forced the editors to exclude some very worthwhile papers, and this is of course a matter of regret.

It is hoped that the next conference in this series will be held in 2009 and will be Groups St Andrews in Bath. As Colin and Edmund will have retired before Groups St Andrews 2009, they look forward to this conference with special interest and anticipate a slightly different role. These Proceedings do, however, mark the first 25 years of Groups St Andrews conferences. In addition to the mathematics, illustrated by these Proceedings, the seven conferences have contained a wide selection of social events. "Groups St Andrews tourism" has taken us to a variety of interesting and scenic venues in Scotland, Ireland, England and Wales. Bus trips have included Kellie Castle, Loch Earn and Loch Tay, Falkland Palace and Hill of Tarvit, Crathes Castle and Deeside, Loch Katrine and the Trossachs, House of Dun, Connemara and Kylemore Abbey, the Burren and the Cliffs of Moher, Tintern Abbey and Welsh Valleys, the Roman Baths in Bath, Salisbury Cathedral, Rufus Stone and the New Forest, Stonehenge, Wells Cathedral and the Cheddar Gorge, Blenheim Palace, Glamis Castle. We have been on boats on Loch Katrine, the Thames, and Galway Bay to the Aran Islands. There have also been: musical events with participants as the musicians, Scottish Country Dance evenings, barn dances, piano recitals, organ recitals, theatre trips, whisky tasting, putting, chess, walks along the Fife Coast, walks round Bath, and walks round Oxford. All these have provided opportunities for relaxation, but also opportunities to continue mathematical discussions. We hope that the cricket match (and balloon trip) postponed from 1997 will take place only 12 years late at Groups St Andrews 2009 in Bath!

Thanks to those authors who have contributed articles to these Proceedings. We have edited their articles to produce some uniformity without, we hope, destroying individual styles. For any inconsistency in, and errors introduced by, our editing we take full responsibility. Our final thanks go to Roger Astley and the rest of the Cambridge University Press team for their assistance and friendly advice throughout the production of these Proceedings.

Colin Campbell
Martyn Quick
Edmund Robertson
Geoff Smith

GROUPS ST ANDREWS 2009

INTRODUCTION

Groups St Andrews 2009 was held in the University of Bath from 1 August to 15 August 2009. This was the eighth in the series of Groups St Andrews group theory conferences organised by Colin Campbell and Edmund Robertson of the University of St Andrews. The first three were held in St Andrews and subsequent conferences held in Galway, Bath and Oxford before returning to St Andrews in 2005 and to Bath in 2009. There were about 200 mathematicians from 30 countries involved in the meeting as well as some family members and partners. The Scientific Organising Committee of Groups St Andrews 2009 was: Colin Campbell (St Andrews), Martyn Quick (St Andrews), Edmund Robertson (St Andrews), Colva Roney-Dougal (St Andrews), Geoff Smith (Bath), Gunnar Traustason (Bath).

The shape of the conference was similar to the previous conferences (with the exception of Groups St Andrews 1981 and 2005) in that the first week was dominated by five series of talks, each surveying an area of rapid contemporary development in group theory and related areas. The invited speakers were Gerhard Hiss (RWTH Aachen), Volodymyr Nekrashevych (Texas A&M), Eamonn O'Brien (Auckland), Mark Sapir (Vanderbilt) and Dan Segal (Oxford). The second week featured three special days, a Cannon/Holt Day, a B H Neumann Day and an Engel Day. The main speakers at these special days included: for the Cannon/Holt Day, George Havas (Queensland), Claas Roever (Galway), Marston Conder (Auckland); for the B H Neumann Day, two of his sons Peter Neumann (Oxford) and Walter Neumann (Columbia, New York) and also Michael Vaughan-Lee (Oxford), Cheryl Praeger (Western Australia), Gilbert Baumslag (CUNY); and, for the Engel Day, Gunnar Traustason (Bath), Olga Macedonska (Katowice) and Patrizia Longobardi (Salerno). Our thanks are due to Charles Leedham-Green (QMWC, London), Roger Bryant (Manchester) and Gunnar Traustason (Bath) for helping organise the programmes for the special days, and to the speakers on these special days.

Each week contained an extensive programme of research seminars and one-hour invited talks. In the evenings throughout the conference, and during the rest periods, there was an extensive social programme. There were two conference outings. The first was to Stonehenge and Salisbury, and the second was to Stourhead Gardens and Wells. In the first week there was a conference banquet at Cumberwell Golf Club. In the second week there was a wine reception at the American Museum in Britain for the B H Neumann Day and a conference banquet at Bath Racecourse on the Cannon/Holt Day. We wish to thank Charles Leedham-Green for allowing us to publish the after-dinner address that he gave at the banquet. Once again the 'Daily Group Theorist' was a nice feature of the conference. We thank the various editors of this, by now traditional, publication.

Once again, we believe that the support of the two main British mathematics societies, the Edinburgh Mathematical Society and the London Mathematical Society has been an important factor in the success of these conferences. As well as supporting some of the expenses of the main speakers, the grants from these societies were used to support postgraduate students and also participants from Scheme 5 and fSU countries.

As has become the tradition, all the main speakers have written substantial articles for these Proceedings. These articles along with the majority of the other papers are of a survey nature. All papers have been subjected to a formal refereeing process comparable to that of a major international journal. Publishing constraints have forced the editors to exclude some very worthwhile papers, and this is of course a matter of regret. Volume 1 begins with the papers by the main speakers Gerhard Hiss and Volodymyr Nekrashevych. These are followed by those papers whose first-named author begins with a letter in the range A to E. Volume 2 begins with the papers by the main speakers Eamonn O'Brien, Mark Sapir and Dan Segal. These are followed by those papers whose first-named author begins with a letter in the range F to Z.

The next conference in this series will be held in St Andrews in 2013. We are confident that this will be, as usual, a chance to meet many old friends and to make many new friends.

We would like to thank Martyn Quick, Colva Roney-Dougal, Geoff Smith and Gunnar Traustason both for their editorial assistance with these Proceedings and for all their hard work in organising the conference. Our final thanks go not only to the authors of the articles but also to Roger Astley and the rest of the Cambridge University Press team for their assistance and friendly advice throughout the production of these Proceedings.

CMC, EFR

GROUPS ST ANDREWS 2013

INTRODUCTION

Groups St Andrews 2013 was held at the University of St Andrews from 3rd August to 11th August 2013. This was the ninth in the series of Groups St Andrews group theory conferences organised by Colin Campbell and Edmund Robertson of the University of St Andrews. There were just under 200 mathematicians from over 20 countries involved in the meeting as well as some family members and partners. The Scientific Organising Committee of Groups St Andrews 2013 (all from St Andrews) was Colin Campbell, Max Neunhöffer, Martyn Quick, Edmund Robertson and Colva Roney-Dougal.

This time the academic business of the conference ran for seven days from Sunday 4th August to Saturday 10th August. Four main speakers delivered four talks each, surveying areas of contemporary development in group theory and related areas; Emmanuel Breuillard (Université Paris Sud 11), Martin Liebeck (Imperial College), Alan Reid (University of Texas), and Karen Vogtmann (Cornell University). There were five invited speakers delivering one-hour plenary talks: Peter Cameron (St Andrews), Radha Kessar (City University, London), Markus Lohrey (Universität Leipzig), Derek Robinson (University of Illinois at Urbana-Champaign) and Christopher Voll (University of Bielefeld). In addition there were nearly 100 contributed short talks from the delegates.

In the evenings throughout the conference there was an extensive social programme. The main conference outing was to the Royal Burgh of Falkland either to visit Falkland Palace or, as it turned out, to go on an adventure walk in the Lomond Hills. Other highlights of the social programme were a whisky tasting evening, a musical evening and the conference banquet. Once again 'The Daily Group Theorist' was a nice feature of the conference. We thank the various editors of this, by now traditional, publication.

The support of the two main United Kingdom mathematics societies, the Edinburgh Mathematical Society and the London Mathematical Society has, once again, been an important factor in the success of these conferences. As well as supporting some of the expenses of the main speakers, the grants from these societies were used to support postgraduate students and also participants from Scheme 5 and fSU countries.

Once again all the main speakers have written substantial articles for these Proceedings. The majority of the other papers are of a survey nature. Regrettably we have been limited to one volume so that, even more than has been the case in the past, we have been forced to exclude many worthwhile papers.

We would like to thank Martyn Quick and Colva Roney-Dougal not only for their editorial assistance with these Proceedings but also, along with Max Neunhöffer, for all their hard work in organising the conference.

CMC, EFR

GROUPS ST ANDREWS 2017

INTRODUCTION

Groups St Andrews 2017 was held at the University of Birmingham from 5th August to 13th August 2017. This was the tenth in the series of Groups St Andrews group theory conferences. There were over 200 mathematicians involved in the meeting as well as some family members and partners. The Scientific Organising Committee of Groups St Andrews 2017 was Colin Campbell, Martyn Quick, Edmund Robertson and Colva Roney-Dougal (all from St Andrews) and Chris Parker (Birmingham).

The academic business of the conference ran for seven days from Sunday 6th August to Saturday 12th August. Four main speakers delivered four talks each, surveying areas of contemporary development in group theory and related areas: Michael Aschbacher (Caltech), Pierre-Emmanuel Caprace (Université Catholique de Louvain), Radha Kessar (City, University of London) and Gunter Malle (TU Kaiserslautern). There were five invited speakers delivering one-hour plenary talks: Tim Burness (University of Bristol), Vincent Guirardel (Université de Rennes 1), Harald Helfgott (University of Göttingen), Andrei Jaikin-Zapirain (Universidad Autónoma de Madrid) and Donna Testerman (École Polytechnique Fédérale de Lausanne). In addition there were about 115 contributed short talks from the delegates.

In the evenings throughout the conference there was an extensive social programme. The main conference outing was a choice between Stratford upon Avon (with a chance to visit buildings and sites associated with William Shakespeare) and Warwick Castle. Other highlights of the social programme were a wine reception, a musical evening and the conference dinner. Once again The Daily Group Theorist was a nice feature of the conference. We thank the various editors of this, by now traditional, publication.

The support of the two main United Kingdom mathematics societies, the Edinburgh Mathematical Society and the London Mathematical Society has, once again, been an important factor in the success of these conferences. As well as supporting some of the expenses of the main speakers, the grants from these societies were used to support postgraduate students and also participants from Scheme 5 countries. Financial support was also received from the Heilbronn Institute for Mathematical Research and the publishers de Gruyter, Elsevier and Springer Nature.

Once again all the main speakers have written substantial surveys, in one case a joint article, for these Proceedings. The other papers are also of a survey nature.

We would like to thank Martyn Quick and Colva Roney-Dougal not only for their editorial assistance with these Proceedings but also for all their hard work in organising the conference. Last, and by no means least, we would like to thank Chris Parker, the local hero, for all his untiring efforts in making Birmingham such a happy venue.

CMC, EFR

A list of social events for each of the conferences is at THIS LINK.

A list of the speakers and titles for each of the conferences is at THIS LINK.

A history of the Groups St Andrews meetings is at THIS LINK.

A paper by C M Campbell about the Groups St Andrews Conferences and the Galway Group meetings is at THIS LINK.

Last Updated May 2019