Bounding and nonbounding finite group actions on surfaces
type: Item
instance of: scholarly article
title: Bounding and nonbounding finite group actions on surfaces
zbMATH Open document ID: 1061.57020
published in: Topology and its Applications
publication date: 2004-08-19T00:00:00Z
review text: In the study of transformation groups on 2-dimensional closed surfaces, a basic problem is which finite group actions on closed surfaces extend to compact 3-manifolds. This is equivalent to asking which finite group actions on closed surfaces bound. The author of this paper considers the problem when the group is a linear fractional group \(\text{PSL}(2, p^n)\) and the action is orientation-preserving. Theorem 1. Let \(p^n\) be a prime power such that either \(p\equiv 1\mod 4\), or \(n\) is even or \(p=2\). Then every action of a linear fractional group \(\text{PSL}(2, p^n)\) on a closed orientable surface bounds. For all other values of \(p^n\), each group \(\text{PSL}(2, p^n)\) admits nonbounding actions on closed connected surfaces. A geometric version of Theorem 1 is also given: Theorem 2. Let \(p^n\) be a prime power such that \(p>3\) and either \(p\equiv 1\mod 4\) or \(n\) is even. Then each indecomposable isometric action of \(\text{PSL}(2, p^n)\) on a hyperbolic surface bounds geometrically. In particular, all Hurwitz and genus actions of such groups bound geometrically. In addition, the author computes the 2-dimensional equivariant bordism group \(\Omega_2(\text{PSL}(2, 7))\). Theorem 3. The 2-dimensional bordism group \(\Omega_2(\text{PSL}(2, 7))\) of the group \(\text{PSL}(2, 7)\) is infinite cyclic, generated by the Hurwitz action of \(\text{PSL}(2, 7)\) on Klein's quartic.
reviewed by: Zhi Lü
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57S17
zbMATH DE Number: 2092799
zbMATH Keywords: finite group action
zbMATH Keywords: linear fractional group
zbMATH Keywords: bounding action
zbMATH Keywords: 2-dimensinal cobordism group
zbMATH Keywords: Isometric group action
author: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: j.topol.2003.10.003
OpenAlex ID: W1991415645
cites work: On Torsion-Free Discrete Subgroups of PSL(2, C) with Compact Orbit Space
cites work: Q4314557
cites work: Representing \(PSL_ 2(p)\) on a Riemann surface of least genus
cites work: The genus of PSl2 (q).
cites work: Extending finite group actions on surfaces to hyperbolic 3-manifolds
cites work: Q5587140
cites work: Group Actions on Handlebodies
cites work: Extending finite group actions from surfaces to handlebodies
cites work: Surfaces and planar discontinuous groups. Revised and expanded transl. from the German by J. Stillwell
cites work: Über Abbildungsklassen von Henkelkörpern
cites work: Hurwitz Groups and Finite Group Actions on Hyperbolic 3-Manifolds
DOI: 10.1016/J.TOPOL.2003.10.003
description: scientific article; zbMATH DE number 2092799