Expander dependence on generating sets

Determine whether there exist finite groups equipped with two generating sets of the same fixed cardinality such that the Cayley graphs formed using one sequence of generating sets constitute an expander family while those formed using the other sequence do not; in particular, determine whether it suffices for the projections of a fixed generating set to the quotients to be generating.

Background

The paper discusses a proposed rigidity question for expander families of Cayley graphs. Although expansion is often formulated using a fixed generating set, it is unresolved whether changing to another generating set of uniformly bounded size can destroy expansion on the same sequence of finite groups.

The question is relevant to the paper because the dynamical extensions are controlled by monodromy elements that may not generate the entire group uniformly across the finite quotients. Establishing or refuting this generating-set independence would clarify when group expansion can be transferred to decay-of-correlation estimates.

References

Lubotszky and Weiss pose the question of whether there are finite groups $\Gamma_q$ and generating sets $S_q,S\prime_q$ of the same constant cardinality and with $\mathrm{Cay}(\Gamma_q,S_q)$ being an expander family but $\mathrm{Cay}(\Gamma_q,S\prime_q)$ failing to be an expander family.

— Relative ($τ$), Expanders, and Decay of Correlations for certain Expanding Maps  (2609.05271 - Dougall, 4 Sep 2026) in Section 1, Introduction