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A new approach to the genus spectra of abelian $p$-groups

Haimiao Chen, Yang Li

Abstract

Given a finite group $G$, the {\it genus spetrum} ${\rm sp}(G)$ of $G$ is the set of integers $g\geq 0$ such that $G$ can act faithfully on an orientable closed surface of genus $g$ by orientation-preserving homeomorphisms. The determination of ${\rm sp}(G)$ is a classical topic and has a long history, but progress is lacked. In this paper, when $G$ is an abelian $p$-group with $p>2$, we propose a new approach to ${\rm sp}(G)$, giving a structural description for ${\rm sp}(G)$ in terms of a function which can be computed in finitely many steps.

A new approach to the genus spectra of abelian $p$-groups

Abstract

Given a finite group , the {\it genus spetrum} of is the set of integers such that can act faithfully on an orientable closed surface of genus by orientation-preserving homeomorphisms. The determination of is a classical topic and has a long history, but progress is lacked. In this paper, when is an abelian -group with , we propose a new approach to , giving a structural description for in terms of a function which can be computed in finitely many steps.
Paper Structure (4 sections, 7 theorems, 56 equations)

This paper contains 4 sections, 7 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Group-theoretic aspect
  3. Combinatorial aspect
  4. Recovering some existing results

Key Result

Theorem 1.4

(a) The reduced genus spectrum is given by (b) The reduced minimum genus $\mu_0(\tau)$ equals (c) Let $\lambda_\ast=\max_{0\le x<p^e}\lambda(x)$, and $\overline{\lambda}_\ast=(np^e+\lambda_\ast)/2$. Then the reduced stable upper genus is given by

Theorems & Definitions (18)

  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.4
  • proof
  • ...and 8 more