Periodic Reporting for period 4 - InvGroGra (Asymptotic invariants of discrete groups, sparse graphs and locally symmetric spaces)
Período documentado: 2020-01-01 hasta 2021-06-30
The main problem raised in the project is to better understand finite objects arising in mathematics through their limiting behavior, and to better understand infinite objects by their finite approximations. In particular, consider limits of discrete structures, like finite graphs or processes on finite graphs and limits of continuous structures, like Riemannian manifolds or the homology and eigenfunctions on these manifolds. The importance of this project for society is that it provides a framework in which one can put an order on the universe of very big finite objects, like networks or structured large data. The overall objectives are to understand specific instances of this limiting theory, including locally symmetric spaces, processes on finite graphs, unimodular random graphs and Riemannian manifolds, invariant random subgroups of discrete and Lie groups and invariant stochastic processes on groups and graphs.
The research group (with various coauthors) worked on several sub-projects of the proposed activity. Some already resulted in accepted papers at prestigeous journals, some are at a preprint stage and some are currently being written up. We choose three main results from this period. The PI (with several coauthors) finished a long paper on the growth of L2-invariants of locally symmetric spaces. It turns out that invariant random subgroups can be effectively used to control Benjamini-Schramm convergence of locally symmetric spaces and their L2 invariants, like the growth of homology and representation multiplicities. The paper was recently accepted at the Annals of Mathematics, the leading journal in mathematics. The PI, Nikolay Nikolov and Tsachik Gelander finished their project on the growth of rank and torsion for right angled groups. It turns out that both the growth of rank and the first homology torsion vanish for an arbitrary sequence of subgroups in any right angled lattice in a higher rank simple Lie group. The paper has been accepted at Duke Math. The PI and Ian Biringer introduced the notion of unimodularity in the realm of Riemannian manifolds, laying the foundations of the theory and proving starting results.
The research group obtained a new understanding of large volume locally symmetric spaces and large graphs and processes on them. Their new approach has already been proved useful and more is expected. Pure research in mathematics rarely has a direct socio-economic impact, however, it often leads to such impact indirectly and on the long run. We expect that the sampling limit language we do research on will prove to be useful outside mathematics as well, including quantum mechanics and statistical physics.