Expanding Generator Sets for Solvable Permutation Groups

Let G = 〈S〉 be a solvable permutation group given as input by the generating set S. I.e. G is a solvable subgroup of the symmetric group Sn. We give a deterministic polynomial-time algorithm that computes an expanding generating set of size Õ(n2) for G. More precisely, given a λ < 1, we can compute a subset T ⊂ G of size Õ(n2) ( 1 λ )O(1) such that the undirected Cayley graph Cay(G,T ) is a λ-spectral expander (the Õ notation suppresses log n factors). In particular, this construction yields ε-bias spaces with improved size bounds for the groups Zd for any constant ε > 0. We also note that for any permutation group G ≤ Sn given by a generating set, in deterministic polynomial time we can compute an ( n λ )O(1) size expanding generating set T , such that Cay(G,T ) is a λ-spectral expander; here the constant in the exponent is large but independent of λ.