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Patterns on Symmetric Riemann Surfaces

Published online by Cambridge University Press:  15 April 2019

By
Adnan Melekoğlu  and
David Singerman
Edited by
C. M. Campbell ,
C. W. Parker ,
M. R. Quick ,
E. F. Robertson  and
C. M. Roney-Dougal
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
C. W. Parker
Affiliation:
University of Birmingham
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
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Summary

We are interested in regular maps on compact symmetric Riemann surfaces. A surface is symmetric if it admits an antiholomorphic involution (symmetry). The fixed-point set of this symmetry is a collection of simple closed curves called mirrors. These mirrors pass through the vertices, edge-centres and face-centres of the map forming a sequence which we call a pattern. Klein in 1879 calculated the pattern for the Riemann surface named after him. Here we discuss the patterns for various families of Riemann surfaces, ending with the Hurwitz surfaces, these admitting 84(g − 1) automorphisms.

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Chapter
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Groups St Andrews 2017 in Birmingham , pp. 443 - 454
Publisher: Cambridge University Press
Print publication year: 2019

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