Invariants of residually finite groups: graphs, groups and dynamics

Marie Curie Fellowship PIEF-GA-2009-235545 - ResFinGroups, Miklos Abert

The project has lasted 1 year, from 01st September 2009 to 31st August2010.

The research work during the addressed period has been very successful. There has been a lot of progress on the proposed directions and new, exciting problems and areas emerged. Out of the concrete problems proposed, more than half have been addressed in depth or made significant progress on. This is generally understood to be a good rate in mathematics. On an asymptotic Stuck-Zimmer theorem for higher rank irreducible lattices, there has been a lot of progress made. First, together with N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault and I. Samet, the researcher proved the following results. Let G be a higher rank simple Lie group.
Then for any sequence of lattices in G with covolume tending to infinity, the quotient manifolds converge to G in the Benjamini-Schramm sense. Using this result, one can prove a uniform Luck approximation result on these lattices, namely one can show that the Betti numbers of the quotient manifolds normalized by the covolume converge to the L2 Betti numbers of G. These results are obviously not true for rank 1, but in that case, they still hold if one looks at sequences of congruence covers of a fixed arithmetic lattice. Even more, one can prove that for good dimensions (far enough from the middle dimension) the corresponding Betti numbers in this case converge to the L2 Betti with exponent less than one. This result comes from an asymptotic Stuck-Zimmer theorem that gives sharper estimates on the ratio of fixed points for congruence Schreier graphs of arithmetic groups.
Problems of this nature have been investigated by number theorists like Sarnak and Xue for normal covering towers of arithmetic manifolds. With Y. Glasner and B. Virag, the researcher proved a strong version of Kesten's theorem on spectral radius. First, they proved that d-regular infinite unimodular random Ramanujan graphs are trees. This implies that any sequence of finite Ramanujan graphs locally converges to the tree. A stronger version of this states that for a Ramanujan graph, the essential girth is at least loglog of the size of the graph. This answers a question of Lubotzky.
Another result concerns invariant random subgroups of finitely generated groups. The authors proved that an invariant random subgroup does not increase the exponent of the probability of return (that can also be expressed as the spectral radius of the quotient Schreier graph) a.s. if and only if the random subgroup is amenable a.s. Together with G. Elek, the researcher investigated hyperfiniteness in the realm of permutation actions. An action is hyperfinite if the Cech-Stone compactification admits a regular hyperfinite Borel probability measure that is invariant under the extended action.
They showed that an action is hyperfinite if and only if there exists a Folner sequence in the corresponding Schreier graph. The researcher has established a rigidity theorem on expander Cayley diagrams as follows. For a Cayley diagram that is a good expander, every almost automorphism (that is, a permutation of the vertices that brings most of the edges to edges) is close to an actual automorphism. The result has a counterpart for graphings of property (T) groups as well. Together with G. Elek, the researcher established a weak containment rigidity theorem for profinite actions.
They showed that if two strongly ergodic profinite actions are weakly equivalent then they are isomorphic. This result has a number of applications, for instance it allowed to answer a question of Lubotzky and Zuk negatively, whether for a sequence of subgroups with propoerty tau, the lattice generated by the sequence also has tau. On the question whether the mod p homology growth exists