Approximate representations of groups

In this thesis, we consider various notions of approximate representations of groups. Loosely speaking, an approximate representation is a map from a group into the unitary operators on a Hilbert space that satisfies the homomorphism equation up to a small error. Maps that are close to actual representations are trivial examples of approximate representations, and a natural question to ask is whether all approximate representations of a given group arise in this way. A group with this property is called stable. In joint work with Lev Glebsky, Alexander Lubotzky and Andreas Thom, we approach the stability question in the setting of local asymptotic representations. We provide sufficient condition in terms of cohomology vanishing for a finitely presented group to be stable. We use this result to provide new examples of groups that are stable with respect to the Frobenius norm, including the first examples of groups that are not Frobenius approximable. In joint work with Narutaka Ozawa and Andreas Thom, we generalize a theorem by Gowers and Hatami about maps with non-vanishing uniformity norm. We use this to prove a very general stability result for uniform epsilon-representations of amenable groups which subsumes results by both Gowers-Hatami and Kazhdan