4:30–5:30 pm Eckhart 202
Title: Uniform stability of high-rank Arithmetic groups
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 \(SL(n,\mathbb{Z}), n>2\). 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. 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).