Abstract

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 matrices) is a small deformation of a true unitary representation.  This extends a result of Kazhdan (1982)  for amenable groups and Burger-Ozawa-Thom (2013) for SL(n,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. to a suitable module implies the above stability. The talk is based on joint work with L. Glebsky, N. Monod, and B. Rangarajan (to appear in Memoirs of the EMS ).