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 $\exp(O_ε(dk))$, whil