On a girth-free variant of the Bourgain-Gamburd machine

Let p be a prime number and A ⊆ SL2(Fp) be a set of matrices. Suppose for simplicity that A = A−1, that is, A is a symmetric set. One can consider the Cayley graph Cay(SL2(Fp), A) (here Cay(SL2(Fp), A) = (SL2(Fp), E) and the set of edges E is defined as (x, y) ∈ E iff y = xa, a ∈ A) of the set A and study the properties of Cay(SL2(Fp), A). It is a fundamental problem to show that A is an expander [13], [10] under some conditions on A. Equivalently, we want to estimate nontrivially the operator norm of all representations of the Fourier transform of the characteristic function of A, i.e., Â(ρ), ρ 6= 1. In [1] (also, see [2]) Bourgain and Gamburd obtained

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