Finite symmetries of surfaces of p-groups of co-class 1

Abstract

The genus spectrum of a finite group [Formula: see text] is a set of integers [Formula: see text] such that [Formula: see text] acts on a closed orientable compact surface [Formula: see text] of genus [Formula: see text] preserving the orientation. In this paper, we complete the full classification of spectrum sets of finite [Formula: see text]-groups of co-class [Formula: see text], where [Formula: see text] is an odd prime. As a consequence, it follows that for any prime [Formula: see text] and a finite [Formula: see text]-group of co-class [Formula: see text] of order [Formula: see text] and exponent [Formula: see text], there are at the most seven genus spectra despite the infinite growth of their isomorphism types along with [Formula: see text].