A structured description of the genus spectrum of abelian $p$-groups

The genus spectrum of a finite group $G$ is the set of all $g$ such that $G$ acts faithfully on a compact Riemann surface of genus $g$. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the abelian $p$-groups. Motivated by the work of Talu for odd primes $p$, we develop a general combinatorial machinery, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of abelian $p$-groups. We have a particular view towards how to generally find the reduced minimum genus in this class of groups, determine the complete genus spectrum for a large subclass of abelian $p$-groups, consisting of those groups in a certain sense having `large' defining invariants, and use this to construct infinitely many counterexamples to Talu's Conjecture, saying that an abelian $p$-group is recoverable from its genus spectrum. Finally, we indicate the effectiveness of our combinatorial approach by applying it to quite a few explicit examples.