Uniform stability of lattices in high-rank semisimple groups

TIFR International Colloquium 2024

Alex Lubotzky (Hebrew University)

Lattices in high-rank semisimple groups enjoy several special properties like super-rigidity, 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 D satisfy: every finite-dimensional unitary “almost-representation” of D (almost w.r.t. to a sub-multiplicative norm on the complex matrics) 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 SL(n, Z), n greater than 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 H2 w.r.t. to a suitable module implies the above stability. The talk is based on a joint work with L. Glebsky, N. Monod, and B. Rangarajan.