Mini Course on Uniform Stability of Higher-Rank Arithmetic Groups

These notes are based on lectures given by Alexander Lubotzky and the author in a mini-course at ICTS Bengaluru as part of the program Zariski Dense Subgroups, Number Theory and Geometric Applications in January 2024, and the results are from a sequence of joint works with Lev Glebsky and Nicolas Monod [20], and Francesco Fournier-Facio [16]. Lattices in higher-rank semisimple groups enjoy a number of rigidity properties like super-rigidity, quasi-isometric rigidity, first-order rigidity, and more. In these lectures, we will add another one: uniform stability. Namely, it will be shown that (most) such lattices \(\Gamma \) satisfy the following: Every finite-dimensional unitary “almost-representation” of \(\Gamma \) (with respect to a sub-multiplicative norm) is a small deformation of a true unitary representation. This extends a result of Kazhdan (1982) for amenable groups and or Burger–Ozawa–Thom (2013) for \(SL(n,Z), n > 2\) . Towards this goal, an elaborate cohomological theory is constructed capturing the obstruction to such stability, and it is shown that the vanishing of second cohomology implies stability in this setting. This cohomology can be roughly thought of as an asymptotic version of bounded cohomology, and sheds light on a question raised in [27] about a possible connection between vanishing of second bounded cohomology and Ulam stability. This machinery is then used to establish uniform stability of lamplighters and Thompson groups, and the main result about lattices in higher rank Lie groups.