Finite Symmetries of surfaces of $p$-groups of co-class 1

TL;DR

This paper investigates the genus spectrum of finite $p$-groups of co-class 1, showing there are at most eight possible spectra for given order and exponent, and classifies groups with a unique stable upper genus.

Contribution

It completes the classification of genus spectra for finite $p$-groups of co-class 1 and identifies groups with a unique stable upper genus.

Findings

At most eight genus spectra for fixed order and exponent.

Classification of groups with a unique stable upper genus.

Finite growth of isomorphism types despite spectrum limitations.

Abstract

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 $e$.