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 $\mathrm{ab}_k(G)$, 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 $\mathrm{rep}_k(G)$, the number of irreducible complex representations of degree at most $k$, to be $\exp(O_ε(dk^2))$. We show that the first bound implies the second, and in fact implies it with $k^2$ replaced by $k\log k$. For this, we prove that every finite $d$-generated group satisfies the inequality $\mathrm{rep}_k(G)\leq\exp((d-1)k\log k+O(dk))\cdot\mathrm{ab}_k(G)$, regardless of its spectral gap. The leading term $(d-1)k\log k$ 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 $\mathrm{GL}_m(\mathbf{C})$, and we show that there are at most $\exp(O(\log^4m))$ of them up to conjugacy and scalars. Wigderson asked whether $k^2$ 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.