Uniform stability of lattices in high-rank

Lattices in high-rank semisimple groups enjoy a number of special properties like superrigidity, quasi-isometric rigidity, first-order rigidity, and more. In this talk, we will add another one: uniform (a.k.a. Ulam) stability. Namely, it will be shown that (most) such lattices $\rm D$ satisfy: every finite-dimensional unitary “almost-representation” of $\rm D$ (almost w.r.t. a sub-multiplicative norm on the complex matrices) is a small deformation of a true unitary representation. This extends a result of Kazhdan (1982) for amenable groups and of Burger–Ozawa–Thom (2013) for ${\rm SL}(n,{\mathbb Z})$, $n>2$. The main technical tool is a new cohomology theory (“asymptotic cohomology”) that is related to bounded cohomology in a similar way to the connection of the last one with ordinary cohomology. The vanishing of $H^2$ w.r.t. a suitable module implies the above stability. The talk is based on a joint work with L. Glebsky, N. Monod, and B. Rangarajan. To appear in the Memoirs of the European Mathematical Society.