Аннотация:
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.
