Lay summary

Sofic groups, by their nature, break frontiers between various areas of pure mathematics (geometric group theory, dynamical systems, operator algebras). This is a very promising class of groups whose recent genesis and modern understanding of geometric, asymptotic, and algebraic structures can generate new examples and theories in analytic properties of groups. It appears to be necessary to develop their study further and investigate connections with geometric group theory, topology, and dynamics.Analysis on discrete infinite groups is establishing itself as a new branch of group theory, with techniques borrowing from metric geometry, operator algebras, and harmonic analysis.Since 1980, Gromov is treating finitely generated groups as metric spaces, and a wealth of results by Gromov and followers show that a large part of the algebraic structure is captured by metric properties. In particular, Gromov introduced several asymptotic invariants of groups, i.e. quasi-isometry invariants, robust with respect to "local" perturbations and depending only on the "large-scale" geometry of the group. Many of these invariants are analytic in nature.The main goal of this project is to work out the precise relations between soficity and several group-theoretical properties which emerged recently, some analytic/algebraic (Haagerup property, C*-exactness, hyperlinearity,...), some metric (word hyperbolicity, Yu's property (A),...). Some of these properties imply positive results towards deep conjectures on group algebras, like the Baum-Connes conjecture, and our second objective is to contribute to some of these conjectures, by providing new classes of groups satisfying them.In this project we will focus on groups having geometric content, as a natural setup for doing analysis: hyperbolic groups, groups acting on trees, groups acting on Hilbert spaces, discrete subgroups of Lie groups.