Final Report Summary - RESFINGROUP (Invariants of residually finite groups: graphs, groups and dynamics)
The researcher proposed to investigate the asymptotic behaviour of invariants on the subgroup lattice of residually finite groups. The proposed activity lies at the crossroads of graph theory, group theory and dynamics. It also has strong connections to certain areas in probability theory and topology, in particular, percolation on transitive graphs and the theory of three-manifolds. The main objective was to further investigate the connections between asymptotic invariants of covering towers, algebraic invariants of residually finite groups and dynamical properties and invariants of profinite actions, with a special emphasis on the phenomenon that unimodular random graphs tend to behave like vertex transitive graphs.
The proposed research was very successful and a lot of new results and directions emerged from it. The four most successful directions were the following.
Together with N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault and I. Samet, the researcher proved that for a higher rank simple real Lie group, for any sequence of lattices with covolume tending to infinity, the quotient manifolds converge to the Lie group in the Benjamini-Schramm sense. This result has a number of applications ranging from the growth of Betti numbers to counting multiplicities of unitary representations. In particular, one can prove that the Betti numbers of the quotient manifolds normalised by the covolume converge to the L2 Betti numbers of the Lie group. For congruence subgroups of arithmetic groups (even in rank one) one can also obtain much stronger results and explicite estimates on the rate of convergence.
The researcher has established a rigidity theorem on expander Cayley diagrams, where he showed that every almost automorphism of a Cayley diagram that is a good expander must be close to a proper automorphism. The result comes from dynamics where rigidity theorems of similar nature were known for measure preserving actions of Kashdan groups.
With Y. Glasner and B. Virag, the researcher proved a strong version of Kesten's theorem on spectral radius, by generalising Kesten's theorem on Ramanujan vertex transitive graphs to unimodular random graphs and by proving the measurable version of Kesten's theorem on how the spectral radius grows when factoring out by a normal subgroup. Together with T. Hubai the researcher investigated the chromatic polynomial of finite graphs and proved that the real moments of the uniform probability measure on the chromatic roots of a finite graphs are convergent for a Benjamini-Schramm convergent sequence of graphs.
This result is connected to questions in statistical physics through the anti-ferromagnetic Ising model. All of the results above open new areas of investigation and as they all connect distinct areas of mathematics, it is expected that they will have further impact besides their intrinsic interest in mathematics.
The proposed research was very successful and a lot of new results and directions emerged from it. The four most successful directions were the following.
Together with N. Bergeron, I. Biringer, T. Gelander, N. Nikolov, J. Raimbault and I. Samet, the researcher proved that for a higher rank simple real Lie group, for any sequence of lattices with covolume tending to infinity, the quotient manifolds converge to the Lie group in the Benjamini-Schramm sense. This result has a number of applications ranging from the growth of Betti numbers to counting multiplicities of unitary representations. In particular, one can prove that the Betti numbers of the quotient manifolds normalised by the covolume converge to the L2 Betti numbers of the Lie group. For congruence subgroups of arithmetic groups (even in rank one) one can also obtain much stronger results and explicite estimates on the rate of convergence.
The researcher has established a rigidity theorem on expander Cayley diagrams, where he showed that every almost automorphism of a Cayley diagram that is a good expander must be close to a proper automorphism. The result comes from dynamics where rigidity theorems of similar nature were known for measure preserving actions of Kashdan groups.
With Y. Glasner and B. Virag, the researcher proved a strong version of Kesten's theorem on spectral radius, by generalising Kesten's theorem on Ramanujan vertex transitive graphs to unimodular random graphs and by proving the measurable version of Kesten's theorem on how the spectral radius grows when factoring out by a normal subgroup. Together with T. Hubai the researcher investigated the chromatic polynomial of finite graphs and proved that the real moments of the uniform probability measure on the chromatic roots of a finite graphs are convergent for a Benjamini-Schramm convergent sequence of graphs.
This result is connected to questions in statistical physics through the anti-ferromagnetic Ising model. All of the results above open new areas of investigation and as they all connect distinct areas of mathematics, it is expected that they will have further impact besides their intrinsic interest in mathematics.