We study approximate homomorphisms into symmetry groups. These are families of maps satisfying the homomorphism condition asymptotically. Stability is the property that every such approximate homomorphism can be replaced by an exact homomorphism close to it. Every approximate homomorphism determines a homomorphism into the universal sofic group. A group is sofic if there exists a monomorphism to the universal sofic group. A group is stable if every sofic morphism is perfect, meaning it filters to an exact homomorphism. If a group is stable and sofic, it is residually finite. A consequence is that groups with property (T) are usually not stable. A sofic representation of a group is a morphism from the group to the universal sofic group that additionally has no fixed points, that is, it has zero trace. There exists a convex structure on the space of sofic representations. In this space, amenable groups are extreme points and (up to amplifications and conjugations) have only one representation. As a consequence, finitely generated abelian groups are stable. Invariant probability measures on the subgroups of a group characterize stability of amenable groups. Specifically, an amenable group is stable if every such invariant measure is co-sofic, i.e. supported on finite-index subgroups.
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