Description of the project

The general subject area is the study of infinite groups, through their intrinsic geometry and the geometry of their actions on various types of metric spaces.

Some of the most remarkable infinite groups (e.g. semisimple Lie groups and their lattices, groups of automorphisms of free groups, mapping class groups) come endowed with actions on spaces with good geometry (symmetric spaces, outer spaces, Teichmüller spaces, complexes of curves etc).

Other interesting actions to consider are those on Hilbert and Banach spaces, and on various non-positively curved spaces (e.g. trees, products of trees,CAT(0) cube complexes etc). For such actions fixed point properties (like Kazhdan's property (T)) and, at the other extreme, existence of proper actions (such as the Haagerup property/ a-(T)-menability) may encode a lot of geometric and analytic information about the group.

Both the fixed point properties and the existence of proper actions may be seen as properties of (non)-existence of good equivariant embeddings into various types of spaces. Through this interpretation these meet themes that have long been of interest in theoretical computer science, i.e. of studying graphs and their (equivariant when acted on by groups) embeddings into various spaces with well understood geometries. Various parameters measuring how well does a (simplicial or even metric; in particular Cayley) graph embed in a metric space in a given collection may encode a lot of information on the graph or group considered. Among such parameters one can count distortion, multiplicative or additive, and compression. Some intrinsic structures, such as metric median, measured walls structure, closely relate to the existence of such embeddings.

The aim of this project is to gather together specialists around these themes, in particular people studying groups from the geometric, analytic or combinatorial point of view and computer scientists. This will be done through a series of meetings, conferences and workshops.

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