Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups

It is by now classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list: uniform stability with respect to the family of unitary operators on finite-dimensional Hilbert spaces equipped with submultiplicative norms. Namely, we show that for (most) high rank lattices, every finite-dimensional unitary “almost-representation” of $\Gamma$ is a small deformation of a (true) unitary representation. This extends a result of Kazhdan (1983) for amenable groups and of Burger–Ozawa–Thom (2013) for $\mathrm{SL}(n,\mathbf{Z})$ (for $n>2$). Towards this goal, we first build an elaborate cohomological theory capturing the obstruction to such stability, and show that the vanishing of the second cohomology implies uniform stability in this setting. This cohomology can be roughly thought of as an asymptotic variant of bounded cohomology, and sheds light on a question raised by Monod (2006) about a possible connection between vanishing of the second bounded cohomology and Ulam stability.