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On one-relator quotients of the modular group

Published online by Cambridge University Press:  05 July 2011

Marston Conder
Affiliation:
University of Auckland, New Zealand
George Havas
Affiliation:
The University of Queensland, Australia
M. F. Newman
Affiliation:
Australian National University, Australia
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

We investigate the modular group as a finitely presented group. It has a large collection of interesting quotients. In 1987 Conder substantially identified the one-relator quotients of the modular group which are defined using representatives of the 300 inequivalent extra relators with length up to 24. We study all such quotients where the extra relator has length up to 36. Up to equivalence, there are 8296 more presentations. We confirm Conder's results and we determine the order of all except five of the quotients. Once we find the order of a finite quotient it is easy to determine detailed structural information about the group. The presentations of the groups whose order we have not been able to determine provide interesting challenge problems.

Our study of one-relator quotients of the modular group is ‘in the small’, that is, with a short extra relator. We briefly compare and contrast our results with generic results.

Introduction

The modular group is a much studied object in mathematics. Indeed in the documentation for the award of the 2009 Abel Prize to Mikhail Gromov, this group is described as “one of the most important groups in the modern history of mathematics”. It is perhaps best known as the projective special linear group L2(ℤ), with a standard representation as a group of linear fractional transformations. It has a large collection of interesting quotients, including most of the nonabelian finite simple groups.

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Publisher: Cambridge University Press
Print publication year: 2011

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