Local randomness and spectral independence

Vortrag von Prof. Dr. Alireza Golsefidy

Datum: 14.04.21  Zeit: 16.00 - 17.15  Raum: ETH HG G 19.2

I will talk about two new concepts related to random walks in compact groups. Roughly locally random compact groups are groups that do not have low complexity models. I will indicate how such a property gives us a mixing inequality. Two compact groups G and H are called spectrally independent if the following holds: suppose (X,Y) is a random variable with values in GxH. If X and Y have spectral gap property in G and H, respectively, then (X,Y) has spectral gap property in GxH. I will explain why compact open subgroups of two non-locally isomorphic almost simple analytic groups are spectrally independent. Along the way we show that a local approximate homomorphism between two compact open subgroups of almost simple analytic groups that has large image is close to an isogeny. Among other things this result extends a work of Kazhdan on approximate homomorphisms. (Joint with K. Mallahi-Karai and A. Mohammadi)