In an ongoing joint work with Glebsky, Lubotzky and Monod, we construct an analogue 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.

In the first of a two-part talk, I will discuss the notion of uniform stability, and the idea of "defect diminishing", which allows us to carry out corrections in a sequence of smaller steps where each step involves a homomorphism lifting problem with abelian kernel. This in turn motivates a cohomological theory (which we call asymptotic cohomology) capturing obstructions to such a lift, so that the vanishing of the second cohomology group in this setting would imply uniform stability.