Asymptotic Bounded Cohomology and Uniform Stability of high-rank lattices

In ongoing joint work with Glebsky, Lubotzky, and Monod, we construct an analog of bounded cohomology in an asymptotic setting in order to prove uniform stability of lattices in Lie groups (of rank at least two) with respect to unitary groups equipped with a metric induced by a submultiplicative norm.  The main idea is the notion of "defect diminishing", which allows us to reduce stability as a homomorphism lifting problem with an abelian kernel, and relates to an asymptotic bounded cohomology H2a whose vanishing implies uniform stability.  The proof of the vanishing of this H2a for lattices in high-rank Lie groups will be on the lines of the corresponding vanishing results for bounded cohomology as discussed in Prof. Monod's talk from last week.