Two basic measures of topological complexity for a 3-manifold are its rank (the number of generators of its fundamental group) and the size of its first homology group. If the 3-manifold is hyperbolic, then another such measure is the first eigenvalue of the Hodge Laplacian on coexact 1-forms (by Mostow rigidity, this is a topological invariant). It is natural to ask how "low complexity" a hyperbolic 3-manifold can be with respect to all of these measures. I will discuss both positive and negative results in this direction. In particular, I will describe (at least) one construction of infinitely many hyperbolic rational homology 3-spheres with coexact 1-form eigenvalues uniformly bounded away from zero. This is joint work with Amina Abdurrahman, Vikram Giri, Ben Lowe, and Jonathan Zung.