Geometric, analytic and algorithmical aspects of groups – GGAA

The idea of understanding a geometry via the group of transformations preserving it, dating back to Felix Klein and the Erlangen program, has gradually come to be used in the reversed order: to try to understand a group via its actions on some (metric) space with a good structure. Several of the most interesting and ubiquitous groups (e.g. lattices, Out(Fn), mapping class groups) act on various ”symmetric-like” spaces. These actions are a rich source of information about the groups, and are the basis of many deep analogies.

The point of view of understanding groups through their actions inspired new equivalence relations and the search for invariants with respect to these relations. Quasi-isometric equivalence was designed partly to cover (though not to be equivalent to) the situation when two groups G1 and G2 act properly discontinuously and with compact quotient on the same metric space X.

One can measure the ‘compatibility’ between a group G and the geometry of a metric space X by studying the metric difference between an orbit and the group itself. For infinite groups a measure of this compatibility is compression, while for finite groups and finite graphs another parameter is distortion.

In this study of graphs and their (equivariant) embeddings into various spaces geometers and analysts meet themes that have arisen much interest in theoretical computer science and combinatorial optimisation. In these areas a way of solving problems consists in embedding the combinatorial structure under consideration into a `well understood metric space' and in using the ambient `good geometry' to devise an algorithm. Within the theme of embeddings into Euclidean and Hilbert spaces two classes of graphs are particularly significant: expander graphs and median graphs.

This project is about all these highly interconnected themes, mixing geometry, analysis and combinatorics.