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A survey on the minimum genus and maximum order problems for bordered Klein surfaces

Published online by Cambridge University Press:  05 July 2011

By
E. Bujalance ,
F. J. Cirre ,
J. J. Etayo ,
G. Gromadzki  and
E. Martínez
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
E. Bujalance
Affiliation:
UNED, Spain
F. J. Cirre
Affiliation:
UNED, Spain
J. J. Etayo
Affiliation:
Universidad Complutense, Spain
G. Gromadzki
Affiliation:
Gdańsk University, Poland
E. Martínez
Affiliation:
UNED, Spain
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

Every finite group acts as a group of automorphisms of some compact bordered Klein surface of algebraic genus g ≥ 2. The same group G may act on different genera and so it is natural to look for the minimum genus on which G acts. This is the minimum genus problem for the group G. On the other hand, for a fixed integer g ≥ 2, there are finitely many abstract groups acting as a group of automorphisms of some compact bordered Klein surface of algebraic genus g. The condition g ≥ 2 assures that all such groups are finite. So it makes sense to look for the largest order of groups G acting on some surface of genus g when g is fixed and G runs over a prescribed family F of groups. This is the maximum order problem for the family F. There is a significant amount of research dealing with these two problems (or with some of their variations), and the corresponding results are scattered in the literature. The purpose of this survey is to gather some of these results, paying special attention to important families of finite groups.

Introduction

A natural extension of the definition of a compact Riemann surface, which is orientable and has no boundary, is to allow dianalytic transition functions, that is, functions which are either analytic or the composite of complex conjugation with an analytic function.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 161 - 182
Publisher: Cambridge University Press
Print publication year: 2011

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