Geometric and analytic aspects of infinite groups

Lead Research Organisation: University of Oxford
Department Name: Mathematical Institute

Abstract

We study infinite groups via their actions on various classes of spaces, with a particular emphasis on two types of actions, in some sense extreme:(a) Actions with a global fixed point. The property (called fixed point property) of a group of having only such actions on spaces in a given class may have strong implications. Kazhdan's property (T) is the most important version of fixed point property. Taking finite quotients of groups with property (T) is one of the most used ways to construct families of expanders. (b) Proper actions. This means on the contrary that only finitely many elements in the group translate a point in a compact set to a point in the same set. In other words, each orbit of the group is a faithful enough picture of the group itself, drawn on the blackboard'' provided by a space in the given collection. Various versions of amenability are connected with such actions.We focus on actions on the following classes of spaces:(1) Hilbert spaces and Banach spaces. Hilbert spaces, which are in some sense infinite dimensional generalisations of the familiar Euclidean spaces, seem the ideal blackboard'' on which to draw an infinite group. Surprisingly enough, a proper embedding of an infinite group in a Hilbert space (more generally in a uniformly convex Banach space) is not granted, its very existence, as well as the parameter called compression measuring how much this embedding distorts the group, encapsulate a lot of information on the group. The Rapid Decay property, an important information on the C-star algebra of the group, relevant to the Novikov and Baum-Connes conjectures via Vincent Lafforgue's work, is also defined in terms of an action (linear this time) of the group on the Hilbert space of square-summable real functions on it.(2) CAT(0) spaces (i.e. non-positively curved spaces, in a metrical sense). Interesting particular cases are the cube complexes (with one-skeleta the median graphs) and their non-discrete generalisations the median spaces, and real trees.(3) Symmetric spaces. The most important actions on such spaces are those ofarithmetic lattices (such as the group of square matrices with integer entries); they have close connections with various Number Theory problems. The understanding of such actions brings valuable information on the geometry of arithmetic lattices, some of the most interesting infinite groups.(4) Actions on limit spaces, appearing as limit actions of groups, in problems of compactification of spaces of representations. These actions relate to several interesting topics mixing group theory and logic: they are used in the recent solution of the Tarski conjecture; the possible number of different limit spaces for a group also relates to the Continuum Hypothesis.

Planned Impact

An underlying theme of the programme is the investigation of graphs and their (equivariant) embeddings into various spaces. These topics come from theoretical computer science and combinatorial optimisation. Some of the problems proposed in the project are formulated by specialists in these areas, and any contribution towards their solution will be valuable and highly interesting to them. In these areas a way of solving problems consists in embedding the combinatorial structure under consideration into a well understood metric space (an Euclidean space, or an infinite dimensional version of it, i.e. a Hilbert space) and in using the ambient good geometry to devise an algorithm. Within this theme two classes of graphs are important: expander graphs ( they represent networks in which information propagates well, and graphs that are hardest to embed into Hilbert spaces) and median graphs (relevant for instance in algorithm design and in optimization theory). The present project should lead to a better understanding of expander and median graphs, ways to construct and embed them into various spaces, their relationship with strong versions of property (T) and with obstructions to embeddings into various spaces. The scientific events gathering together pure and applied mathematicians and computer scientists around problems of exceptional classes of graphs and their embeddings into various spaces multiplied over the past few years. The Bernoulli Centre of the Ecole Polytechnique Federale de Lausanne hosted in January-June 2007 a research program entitled Limits of graphs in group theory and computer science having as main themes embeddings of groups and graphs into Hilbert and Banach spaces, explicit constructions of exceptional graphs, applications in coding theory. The PI and the CI attended conferences and workshops within this semester, and gave mini-courses and research talks. Two similar events took place this year only, in Orleans, France, and in the international conference centre of ETH Zurich located in Ascona. The PI and the CI will continue their activities in this direction, and together with the PDRA and the GS they will participate in events belonging to this trend of knowledge exchange at the interface of several areas of research (e.g. the semester on Discrete Analysis to be held at the Isaac Newton Institute for Mathematical Sciences, January - July 2011, events organised by DIMACS and other centres of discrete mathematics, events announced on DMANET, an electronic news and research network for discrete mathematics and algorithms). They will give talks for the academic community in conferences and seminars in UK and abroad. The weekly workshop run throughout the programme, the supervision of the PI, attendance of junior seminars in Oxford, will help the PDRA and the GS acquire the necessary communication skills. The results obtained will be written as papers which will be circulated, sent to possible users in other areas, sent to preprint servers with a large audience such as the Mathematics ArXiv and CO Combinatorics, (Front for the Mathematics ArXiv of Univ. of California, Davis), the Oxford Mathematical Institute preprint server, DMANET, published in journals.

Publications

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Alessandro Sisto (Author) (2011) Exponential triples in The Electronic Journal of Combinatorics

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Arzhantseva G (2019) Geometry of Infinitely Presented Small Cancellation Groups and Quasi-homomorphisms in Canadian Journal of Mathematics

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Behrstock J (2011) Addendum: Median structures on asymptotic cones and homomorphisms into mapping class groups in Proceedings of the London Mathematical Society

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Behrstock J (2019) Combinatorial higher dimensional isoperimetry and divergence in Journal of Topology and Analysis

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Behrstock J (2011) Median structures on asymptotic cones and homomorphisms into mapping class groups in Proceedings of the London Mathematical Society

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Behrstock, J. (2014) Divergence, thick groups, and short conjugators in Illinois J. Math.

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Bridson M (2015) The virtual first Betti number of soluble groups in Pacific Journal of Mathematics

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Bridson M (2016) Determining Fuchsian groups by their finite quotients in Israel Journal of Mathematics

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Bridson M (2015) The triviality problem for profinite completions in Inventiones mathematicae

 
Description The participants in the grant developed topological and analytical tools and results relevant to mainstream research currently carried out in Geometry, Topology and Algebra.

A main technique in the field, shared with (and inspired by) combinatorics and computer science, is to study whether a given infinite algebraic structure can be embedded into a ``good space'', be it a product of (simplicial) trees, a Hilbert (or Banach space), a manifold with non-positive curvature or a metric generalisation of such a manifold etc. Many results obtained by the participants in this project shed a new light and opened new directions of research in this area. In particular, questions related to fixed point properties in Banach spaces and their connections to the classical Kazhdan property, and to random graphs and groups have been answered. A byproduct of results obtained is that random graphs were proved to be expanders in a strong sense (an expander may be interpreted as a graph representing a network that is robust, i.e. difficult to disconnect). There has been consistent progress concerning the behaviour of random groups in terms of the geometry of the boundary and the existence of a wall structure, and of proper actions on Banach spaces. The relevance of the profinite completion in decidability questions has been elucidated. The embeddability of important groups (e.g. mapping class groups, fundamental groups of 3-manifolds, relatively hyperbolic groups) into products of trees, Banach spaces, non-positively curved spaces has been understood in depth. A new weak version of the property of Gromov hyperbolicity, called acylindrical hyperbolicity, has been analysed in depth and proved very fruitful in many settings.
Exploitation Route Many of our tools are very effective, many of our results open different paths or viewpoints for research. A number of our findings have obvious connections to theoretical computer science, combinatorics and optimisation theory, and answer questions coming from that area.
Sectors Digital/Communication/Information Technologies (including Software),Other

URL http://people.maths.ox.ac.uk/drutu
 
Description They have been used in courses and in textbooks.
Sector Digital/Communication/Information Technologies (including Software),Education
Impact Types Societal

 
Description Chair of the EMS/EWM Scientific Committee
Geographic Reach Europe 
Policy Influence Type Participation in a advisory committee
URL https://womenandmath.wordpress.com/emsewm-scientific-committee/
 
Description Taught Course Centre Course
Geographic Reach National 
Policy Influence Type Influenced training of practitioners or researchers
Impact The TCC course held in Oxford offered training to audience from the 5 universities involved in this system (Bath, Bristol, Imperial, Oxford, Warwick). The course covered some of the topics of the grant for the benefit of graduate students and researchers in the five above mentioned mathematics departments.
URL http://people.maths.ox.ac.uk/drutu/
 
Description Taught Course Centre Course
Geographic Reach National 
Policy Influence Type Influenced training of practitioners or researchers
Impact The TCC course held in Oxford offered training to audience from the 5 universities involved in this system (Bath, Bristol, Imperial, Oxford, Warwick). The course covered some of the topics of the grant for the benefit of graduate students and researchers in the five above mentioned mathematics departments.
URL http://people.maths.ox.ac.uk/drutu/
 
Description Taught Course Centre Course
Geographic Reach National 
Policy Influence Type Influenced training of practitioners or researchers
Impact The course covered some of the topics of the grant for the benefit of graduate students and researchers in five mathematics departments.
URL http://people.maths.ox.ac.uk/drutu/
 
Description Taught Course center annual Course
Geographic Reach National 
Policy Influence Type Influenced training of practitioners or researchers
Impact The annual TCC course that is now held in Oxford offers training in this new direction of research to audience from the 5 universities involved in this system (Bath, Bristol, Imperial, Oxford, Warwick). The course in 2011 covered topics at the core of the grant.
URL http://people.maths.ox.ac.uk/drutu/
 
Description ANR Blanc
Amount € 175,000 (EUR)
Funding ID ANR Blanc ANR-10-BLAN 0116, acronym GGAA 
Organisation National Agency for Research 
Sector Public
Country France
Start 10/2010 
End 10/2015
 
Description CNRS-OXFORD
Amount £10,000 (GBP)
Organisation University of Oxford 
Department John Fell Fund
Sector Academic/University
Country United Kingdom
Start 01/2012 
End 06/2012
 
Description LMS Scheme 2 Award
Amount £900 (GBP)
Funding ID 21111 - LMS Grant Award 
Organisation London Mathematical Society 
Sector Academic/University
Country United Kingdom
Start 01/2012 
End 02/2012
 
Description CNRS-Oxford Collaboration Scheme 
Organisation National Center for Scientific Research (Centre National de la Recherche Scientifique CNRS)
Country France 
Sector Academic/University 
PI Contribution participation to a conference as speakers and attendants
Collaborator Contribution participation to a conference as speakers and attendants
Impact research collaborations, research visits
Start Year 2012
 
Description European Center for Mathematics, Physics and their Interactions 
Organisation University of Lille
Country France 
Sector Academic/University 
PI Contribution participation to workshops, organizing of conferences
Collaborator Contribution participation to workshops, organizing of conferences
Impact exchange of information, series of lectures for graduate students
Start Year 2012
 
Description INI Programme 
Organisation Isaac Newton Institute for Mathematical Sciences
Country United Kingdom 
Sector Academic/University 
PI Contribution The PI was one of the co-organisers of this program. The co-PI, the Visiting Researcher K. Vogtmann, the former PDRA John Mackay and the former PhD student Alessandro Sisto were all long term participants and speakers in workshops and in the weekly seminar.
Collaborator Contribution Isaac Newton Institute provided logistics and an environment of the highest level for the organisation of a top-level programme at the cutting edge of the research in the field.
Impact Dissemination of the results obtained, further promotions of the problems addressed by this grant and hopefully further progress.
Start Year 2013
 
Description MSRI Programme Geometric group theory 
Organisation Mathematical Sciences Research Institute
Country United States 
Sector Charity/Non Profit 
PI Contribution The PI co-organised the programme, together with one of the Visiting Researchers of the EPSRC grant, and the co-PI organised one of the workshops and was a long-term participant throughout the programme. Likewise the PDRA (John MackKay) and the former PhD student (Alessandro Sisto) were long-term participants in the programme.
Collaborator Contribution lectures, conferences, workshops, exchange of information
Impact Exchange of information, opportunity to disseminate the outcomes of the present grant.
Start Year 2012
 
Description Participation in an activity, workshop or similar - co-organisation of one of the five workshops held during the Isaac Newton Institute programme 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact The former PDRA of the grant (John Mackay) co-organised this satellite workshop at the University of Oxford. This is one in a series of workshops entitled ``Young Geometric Group Theory workshops'' held annually, usually with a participation of 100-150 graduate and post-doctoral students.
The one at Oxford had over 170 registered participants, and many more who actually attended the talks. It has been considered a great success.
Year(s) Of Engagement Activity 2017
URL https://www.newton.ac.uk/event/npcw02
 
Description British Mathematical Colloquium 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Professional Practitioners
Results and Impact The British Mathematical Colloquium is the largest pure mathematical conference to be held annually in the UK. It has been held every year since 1949. Its purpose is to disseminate relevant advances in mathematics to a larger audience.
Year(s) Of Engagement Activity 2012
URL https://www.kent.ac.uk/smsas/events/160412.html
 
Description EMS/EWM Scientific Committee 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? Yes
Geographic Reach International
Primary Audience Public/other audiences
Results and Impact A Scientific Committee, consisting of twelve leading women mathematicians, has been established jointly by the European Mathematical Society (EMS) and the association European Women in Mathematics.. Awarding Body - European Mathematical Society and European Women in Mathematics, Name of Scheme - Scientific Committee

A Summer School every two years at the Mittag Leffler Institute, the choice of the EMS Lecturer every two years, organization of General Meetings of the EWM, organization of EWM sattelite meetings of ECM.
Year(s) Of Engagement Activity 2012
 
Description Participation in an activity, workshop or similar - co-organisation of one of the five workshops held during the Isaac Newton Institute programme 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact This workshop was the launching event for the semester-long programme on "Non-Positive Curvature, Group Actions, and Cohomology" held at the Isaac Newton Institute in Cambridge. The programme covered the topics of the grant, and the workshop focused on the strong connections between fundamental properties of groups acting on non-positively curved spaces and geometric properties of those spaces. Its goal has been to give an overview of recent developments in the area, encompassing several different notions of nonpositive curvature and their applications within geometric group theory, geometry, and topology.
Its impact as a launching event has been considerable, for the participants in the workshop, for the long-term participants in the programme and for the research groups working in related areas located in Cambridge.
Year(s) Of Engagement Activity 2017
URL https://www.newton.ac.uk/event/npcw01
 
Description Prospects in Mathematics 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Undergraduate students
Results and Impact LMS Prospects in Mathematics Meetings are annual events for Finalist Mathematics Undergraduates who are considering apply for a PhD after they have completed their current studies.

The meetings feature speakers from a wide range of mathematical fields across the UK who discuss their current research and what opportunities are available to prospective PhD students.
Year(s) Of Engagement Activity 2011
URL http://www.maths.bris.ac.uk/~maxcu/Prospects2011.html
 
Description Stockholm bi-annual summer school 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact member of the Oversight Committee for a bi-annual summer school to be held at the Mittag-Leffler Institut in Stockholm
Year(s) Of Engagement Activity 2014
URL http://www.math.ucsd.edu/~alina/ewm/
 
Description Two plenary talks at the Mathematical Sciences Research Institute 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact The program held at the Mathematical Sciences Research Institute in Berkeley, from August to December 2016, in the format of a `jumbo program' (i.e. no parallel programs) comprised 4 workshops, one of which was introductory. I have been invited to give a plenary talk of 90 minutes at the introductory workshop, on the topics covered by this grant.
I have also been invited to give a plenary talk at one of the other workshops.
Year(s) Of Engagement Activity 2016
URL https://www.msri.org/programs/278
 
Description co-organisation of one of the four workshops held during the Mathematical sciences Research Institute programme on the topic 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact In 2016, August to December, a programme has taken place at MSRI, Berkeley, California, with the broad theme `Geometric Group Theory', under the format `jumbo', i.e. with no parallel program. This program comprised one introductory workshop and three topical workshops. I have been the lead organizer of one of the three topical workshops.
Year(s) Of Engagement Activity 2016
URL https://www.msri.org/workshops/770