Uniform Stability of High-Rank Arithmetic Groups

Explore the uniform stability properties of high-rank arithmetic groups in this advanced mathematics lecture delivered by Alex Lubotzky from the Weizmann Institute. Discover how lattices in high-rank semisimple groups possess special properties including super-rigidity, quasi-isometric rigidity, and first-order rigidity, while learning about the addition of uniform (Ulam) stability to this collection of remarkable characteristics. Examine the proof that most such lattices satisfy the condition that every finite-dimensional unitary "almost-representation" is a small deformation of a true unitary representation, extending previous results by Kazhdan (1982) for amenable groups and Burger-Ozawa-Thom (2013) for SL(n,Z) where n > 2. Delve into the main technical innovation of asymptotic cohomology, a new cohomology theory that relates to bounded cohomology similarly to how bounded cohomology connects to ordinary cohomology. Understand how the vanishing of H² with respect to suitable modules implies stability results, and gain insights from collaborative research with L. Glebsky, N. Monod, and B. Rangarajan that is set to appear in Memoirs of the EMS.