The proposed conference aims to bring together experts from different fields: group theory, ergodic theory, probability theory and descriptive set theory. The common interest is to study properties and invariants of infinite groups using various tools, like spectral theory, L2 theory, entropy, orbit equivalence, profinite methods, random walks and graph convergence. There is already considerable interaction between these fields and the proposed participants, but much more is needed. The conference will allow participants to assimilate each others language and will open new connections.