# The genera, reflexibility and simplicity of regular maps

• Volume: 012, Issue: 2, page 343-364
• ISSN: 1435-9855

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## Abstract

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This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to $g-1$, where $g$ is the genus, all orientably-regular maps of genus $p+1$ for $p$ prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and that orientable surfaces of infinitely many genera carry no reflexible regular map with simple underlying graph. Another consequence is a simpler proof of the Breda–Nedela–Širáň classification of non-orientable regular maps of Euler characteristic $-p$ where $p$ is prime.

## How to cite

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Conder, Marston, Širáň, Jozef, and Tucker, Thomas. "The genera, reflexibility and simplicity of regular maps." Journal of the European Mathematical Society 012.2 (2010): 343-364. <http://eudml.org/doc/277173>.

@article{Conder2010,
abstract = {This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to $g-1$, where $g$ is the genus, all orientably-regular maps of genus $p+1$ for $p$ prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and that orientable surfaces of infinitely many genera carry no reflexible regular map with simple underlying graph. Another consequence is a simpler proof of the Breda–Nedela–Širáň classification of non-orientable regular maps of Euler characteristic $-p$ where $p$ is prime.},
author = {Conder, Marston, Širáň, Jozef, Tucker, Thomas},
journal = {Journal of the European Mathematical Society},
keywords = {regular map; symmetric graph; embedding; genus; chiral; reflexible; regular map; symmetric graph; embedding; genus; chiral; reflexible},
language = {eng},
number = {2},
pages = {343-364},
publisher = {European Mathematical Society Publishing House},
title = {The genera, reflexibility and simplicity of regular maps},
url = {http://eudml.org/doc/277173},
volume = {012},
year = {2010},
}

TY - JOUR
AU - Conder, Marston
AU - Širáň, Jozef
AU - Tucker, Thomas
TI - The genera, reflexibility and simplicity of regular maps
JO - Journal of the European Mathematical Society
PY - 2010
PB - European Mathematical Society Publishing House
VL - 012
IS - 2
SP - 343
EP - 364
AB - This paper uses combinatorial group theory to help answer some long-standing questions about the genera of orientable surfaces that carry particular kinds of regular maps. By classifying all orientably-regular maps whose automorphism group has order coprime to $g-1$, where $g$ is the genus, all orientably-regular maps of genus $p+1$ for $p$ prime are determined. As a consequence, it is shown that orientable surfaces of infinitely many genera carry no regular map that is chiral (irreflexible), and that orientable surfaces of infinitely many genera carry no reflexible regular map with simple underlying graph. Another consequence is a simpler proof of the Breda–Nedela–Širáň classification of non-orientable regular maps of Euler characteristic $-p$ where $p$ is prime.
LA - eng
KW - regular map; symmetric graph; embedding; genus; chiral; reflexible; regular map; symmetric graph; embedding; genus; chiral; reflexible
UR - http://eudml.org/doc/277173
ER -

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