  In this note, we lay the groundwork for a new approach to the problem of
group-signature classification of group actions on closed Riemann surfaces.
This new approach first focuses on analyzing the low level arithmetic
conditions on signatures before invoking the more complicated group theory. We
provide the complete first step in this approach by giving the complete list of
signatures which arithmetically could appear as a signature in every possible
genus, and the subset of those which do appear as the signature of a group
action in every possible genus.
