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.
Organisations
- University of Oxford (Lead Research Organisation)
- National Center for Scientific Research (Centre National de la Recherche Scientifique CNRS) (Collaboration)
- Mathematical Sciences Research Institute (Collaboration)
- Isaac Newton Institute for Mathematical Sciences (Collaboration)
- University of Lille (Collaboration)
Publications
Behrstock J
(2019)
Combinatorial higher dimensional isoperimetry and divergence
in Journal of Topology and Analysis
Drutu C
(2019)
Random groups, random graphs and eigenvalues of p-Laplacians
in Advances in Mathematics
Drutu C
(2019)
Kazhdan projections, random walks and ergodic theorems
in Journal für die reine und angewandte Mathematik (Crelles Journal)
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 guidance/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 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 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 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 |