  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.
