Approximate Representations and Approximate Homomorphisms

Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups: functions ψ : G → Ud such that Pr[ψ(xy) = ψ(x) ψ(y)] is large, or more generally Ex,y ‖ψ(xy)− ψ(x) ψ(y)‖22 is small, where x, y are uniformly random elements of the group G and Ud denotes the unitary group of degree d. We bound these quantities in terms of the ratio d/dmin where dmin is the dimension of the smallest nontrivial representation of G. As an application, we bound the extent to which a function f : G → H can be an approximate homomorphism where H is another finite group. We show that if H’s representations are significantly smaller than G’s, no such f can be much more homomorphic than a random function. We interpret these results as showing that if G is quasirandom, that is, if dmin is large, then G cannot be embedded in a small number of dimensions, or in a less-quasirandom group, without significant distortion of G’s multiplicative structure. We also prove that our bounds are tight by showing that minors of genuine representations and their polar decompositions are essentially optimal approximate representations. In additive combinatorics and number theory, an approximate subgroup of a group G is a subset H which is roughly closed under multiplication: that is, such that Prx,y[xy ∈ H] is large where x, y are uniformly random elements of H. We focus on approximate group representations—functions ψ from G to Ud, the group of d× d unitary matrices, such that ψ acts roughly like a homomorphism. We then use our results to bound the existence of approximate homomorphisms from G to another finite group H. Let G be a finite group and let ψ : G → Udψ . If ψ(xy) = ψ(x)ψ(y) for all x, y ∈ G, then we call ψ a representation. We are interested in understanding how close ψ can be to a representation if G does not in fact have any dψ-dimensional representations—in particular, in the case where G is quasirandom [1] in the sense that its smallest nontrivial representation has dimension dmin > dψ. We canmeasure the extent to which ψ fails to act as a representation by the expected l2 distance between ψ(xy) and ψ(x)ψ(y), where x and y are chosen uniformly from G. To control the trivial case where ψ(x) = 1 for all x, we assume that Ex ψ(x) is bounded in its operator norm. We also