Finite Symmetries of surfaces of $p$-groups of co-class 1
The genus spectrum of a finite group $G$ is a set of integers $g \geq 2$ such that $G$ acts on a closed orientable compact surface $\Sigma_g$ of genus $g$ preserving the orientation. In this paper we complete the study of spectrum sets of finite $p$-groups of co-class $1$, where $p$ is an odd prime. As a consequence we prove that given an order $p^n$ and exponent $p^e$, there are at the most eight genus spectrum despite the infinite growth of their isomorphism types along $(n,e)$. Based on these results we also classify these groups which has unique stable upper genus $\sigma_e(p^e) - p^e$, where $\sigma_e(p)$ is a constant that depends on $p$ and $e$.