Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups

Abstract

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, namely uniform stability with respect to the family of unitary operators on finite-dimensional Hilbert spaces equipped with submultiplicative norms. Towards this goal, we first build an elaborate cohomological theory capturing the obstruction to such stability, and show that the vanishing of second cohomology implies uniform stability in this setting. This cohomology can be roughly thought of as an asymptotic version of bounded cohomology, and sheds light on a question raised in [Mon06] about a possible connection between vanishing of second bounded cohomology and Ulam stability. Along the way, we use this criterion to provide a short conceptual (re)proof of the classical result of Kazhdan [Kaz82] that discrete amenable groups are Ulam stable. We then use this machinery to establish our main result, that lattices in a class of higher rank semisimple groups (which are known to have vanishing bounded cohomology) are uniformly stable.

Bounded cohomology is not a profinite invariant

“…It is a straightforward consequence of the Burger-Monod theorem that lattices in higher-rank linear Lie groups are uniformly U(1)-stable [10,Theorem 1.0.11]. Most recently, Glebsky, Lubotzky, Monod, and Rangarajan proved the far-reaching generalization that lattices in many higher-rank semisimple Lie groups G are Ulam stable [10, Theorem 0.0.5].…”