Representation and abelianization growth, particularly along expanding sequences

We study two obstructions to a finite group G having a generating d-tuple whose Cayley graph has spectral gap at least $ε$. Lubotzky and Weiss proved that such a gap forces abₖ(G), the largest order of the abelianization of a subgroup of index at most k, to be exp(O_ε(dk)), while de la Harpe, Robertson and Valette proved that it forces repₖ(G), the number of irreducible complex representations of degree at most k, to be exp(O_ε(dk²)). We show that the first bound implies the second, and in fact implies it with k² replaced by klog k. For this, we prove that every finite d-generated group satisfies the inequality repₖ(G)≤exp((d-1)klog k+O(dk))ₖ(G), regardless of its spectral gap. The leading term (d-1)klog k is attained by powers of alternating groups, and it disappears for groups without large alternating sections. The proof reduces to counting the finite primitive subgroups of GLₘ(C), and we show that there are at most exp(O(log⁴m)) of them up to conjugacy and scalars. Wigderson asked whether k² in the second bound can be lowered to k, so our result answers his question up to a logarithm. We do not know whether the logarithm is actually needed. For powers of alternating groups, the natural candidates, we show that a spectral gap keeps the number of factors a fixed power below the maximum allowed by d-generation, and that random generators have spectral gap tending to $0$ already for superpolynomially many factors.