Discrete groups of intermediate rank

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The theme of this project is a recently discovered family of infinite finitely presented discrete groups called the groups of intermediate rank. These groups are at the frontier of two well---known and actively studied classes of discrete groups: hyperbolic groups, in the sense of Gromov, and lattices in Lie groups, which act on symmetric spaces of non compact type, or Euclidian Tits buildings. In this context, one often observes a sharp distinction between rank 1 and higher rank spaces; groups of intermediate rank interpolate between these two worlds. The proposal is centered around three main themes: ergodic group theory, which studies groups through their probability measure preserving actions on standard spaces, geometric group theory, which relies more on geometric properties, such as nonpositive curvature, and operator algebras, which uses the techniques of infinite dimensional analysis. There has been a continuous and long--lasting interplay between these subjects, one of most prominent example being perhaps the Mostow----Margulis----Zimmer rigidity theory. The connections with operator algebras started with the work of von Neumann on dynamical systems and operator algebras in the 40s, and the study of unitary representations and (group) C*-algebras. The Haagerup property, for example, has played a leading role in the developments of groups of intermediate rank. Some of the most important tools, since the first discoveries on these groups around 10 years ago, have come from connections between these fields. The principal investigator, together with his co-authors and students, will further these connections; rank interpolation is, first and foremost, a new domain that needs to be explored and charted. Explicit open problems and new directions are detailed in the proposal. Additionally, the PI will give introductory lectures and write lecture notes aimed at advanced students. He will also work to foster the development of long term international relationships strengthening the existing ties.