Geometric and analytic aspects of infinite groups

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.

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.