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On one-relator quotients of the modular group

Published online by Cambridge University Press:  05 July 2011

By
Marston Conder ,
George Havas  and
M. F. Newman
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
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.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 183 - 197
Publisher: Cambridge University Press
Print publication year: 2011

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References

[1] Selman, Akbulut and Robion, Kirby, A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture and the Andrews–Curtis conjecture, Topology 24 (1985), 375–390.Google Scholar
[2] Gilbert, Baumslag, John W., Morgan and Peter B., Shalen, Generalized triangle groups, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 1, 25–31.Google Scholar
[3] Wieb, Bosma, John, Cannon and Catherine, Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput.24 (1997), 235–265; See also http://magma.maths.usyd.edu.au/magma/
[4] Colin M., Campbell, George, Havas, Alexander, Hulpke and Edmund F., Robertson, Efficient simple groups, Comm. Algebra 31 (2003), 5191–5197.Google Scholar
[5] Colin M., Campbell, George, Havas, Colin, Ramsay and Edmund F., Robertson, Nice efficient presentations for all small simple groups and their covers, LMS J. Comput. Math. 7 (2004), 266–283.Google Scholar
[6] Colin M., Campbell, George, Havas, Colin, Ramsay and Edmund F., Robertson, On the efficiency of the simple groups with order less than a million and their covers, Experiment. Math. 16 (2007), 347–358.Google Scholar
[7] C.M., Campbell, P.M., Heggie, E.F., Robertson and R.M., Thomas, Finite one-relator products of two cyclic groups with the relator of arbitrary length, J. Austral. Math. Soc. Ser. A 53 (1992), no. 3, 352–368.Google Scholar
[8] C.M., Campbell and E.F., Robertson, A deficiency zero presentation for SL(2, p), Bull. London Math. Soc. 12 (1980), no. 1, 17–20.Google Scholar
[9] Colin M., Campbell and Edmund F., Robertson, The efficiency of simple groups of order < 105, Comm. Algebra 10 (1982), no. 2, 217–225.Google Scholar
[10] Colin M., Campbell and Edmund F., Robertson, Presentations for the simple groups G, 105 < |G| < 106, Comm. Algebra 12 (1984), no. 21–22, 2643–2663.Google Scholar
[11] Marston, Conder, Three-relator quotients of the modular group, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 152, 427–447.Google Scholar
[12] Marston, Conder, A surprising isomorphism, J. Algebra 129 (1990), no. 2, 494–501.Google Scholar
[13] Marston, Conder, George, Havas and M.F., Newman, On one-relator quotients of the modular group; supplementary materials, (2009), http://www.itee.uq.edu.au/~havas/orqmg
[14] H.S.M., Coxeter, The abstract groups Gm,n,p, Trans. Amer. Math. Soc. 45 (1939), no. 1, 73–150.Google Scholar
[15] M., Edjvet and A., Juhàsz, The groups Gm,n,p, J. Algebra 319 (2008), no. 1, 248–266.Google Scholar
[16] Anna, Fabianska, PSL, (2009), http://wwwb.math.rwth-aachen.de/~fabianska/PSLHomepage/
[17] The GAP Group, Aachen, GAP – Groups, Algorithms, and Programming, Version 4.4, (2008). See also http://www.gap-system.org
[18] William Rowan, Hamilton, Memorandum respecting a new system of roots of unity, Philos. Mag. 12 (1856), p. 446.Google Scholar
[19] George, Havas and Derek F., Holt, On Coxeter's families of group presentations, submitted (2010).
[20] George, Havas, M.F., Newman, Alice C., Niemeyer and Charles C., Sims, Groups with exponent six, Comm. Algebra 27 (1999), 3619–3638.Google Scholar
[21] George, Havas, Derek F., Holt, P.E., Kenne and Sarah, Rees, Some challenging group presentations, J. Austral. Math. Soc. Ser. A 67 (1999), 206–213.Google Scholar
[22] George, Havas and Colin, Ramsay, Proving a group trivial made easy: a case study in coset enumeration, Bull. Austral. Math. Soc. 62 (2000), no. 1, 105–118.Google Scholar
[23] George, Havas and Colin, Ramsay, Experiments in coset enumeration, in Groups and Computation III, Ohio State University Mathematical Research Institute Publications 8 (de Gruyter, 2001), 183–192.Google Scholar
[24] G., Havas and C., Ramsay. Coset enumeration: ACE version 3.001 (2001). Available as http://www.itee.uq.edu.au/~havas/ace3001.tar.gz
[25] Derek F., Holt, The Warwick automatic groups software, Geometrical and Computational Perspectives on Infinite Groups (ed. Gilbert, Baumslag et al), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 25 (1996), 69–82.Google Scholar
[26] Derek F., Holt and Sarah, Rees, Computing with abelian sections of finitely presented groups, J. Algebra 214 (1999), 714–728.Google Scholar
[27] J., Howie, V., Metaftsis and R.M., Thomas, Finite generalized triangle groups, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3613–3623.Google Scholar
[28] A., Jamali and E.F., Robertson, Efficient presentations for certain simple groups, Comm. Algebra 17 (1989), 2521–2528.Google Scholar
[29] Ilya, Kapovich and Paul E., Schupp, Random quotients of the modular group are rigid and essentially incompressible, J. Reine Angew. Math. 628 (2009), 91–119.Google Scholar
[30] P.E., Kenne, Efficient presentations for three simple groups, Comm. Algebra 14 (1986), no. 5, 797–800.Google Scholar
[31] L., Lèvai, G., Rosenberger and B., Souvignier, All finite generalized triangle groups, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3625–3627.Google Scholar
[32] Vasileios, Metaftsis and Izumi, Miyamoto, One-relator products of two groups of order three with short relators, Kyushu J. Math. 52 (1998), no. 1, 81–97.Google Scholar
[33] G.A., Miller, On the groups generated by two operators, Bull. Amer. Math. Soc. 7 (1901), no. 10, 424–426.Google Scholar
[34] G.A., Miller, Groups defined by the orders of two generators and the order of their product, Amer. J. Math. 24 (1902), no. 1, 96–100.Google Scholar
[35] W., Plesken and A., Fabianska, An L2-quotient algorithm for finitely presented groups, J. Algebra 322 (2009), no. 3, 914–935.Google Scholar
[36] Derek J.S., Robinson, A Course in the Theory of Groups, Second Edition, Graduate Texts Math. 80 (Springer-Verlag, New York 1996).Google Scholar
[37] Paul E., Schupp, Embeddings into simple groups, J. London Math. Soc. (2) 13 (1976), no. 1, 90–94.Google Scholar
[38] C.C., Sims, Computation with finitely presented groups, Encyclopedia of Mathematics and its Applications 48, (Cambridge University Press, 1994).Google Scholar
[39] N.J.A., Sloane. The On-Line Encyclopedia of Integer Sequences, (2009), http://www.research.att.com/~njas/sequences/A000011
[40] Yücel Türker, Ulutaş and Ismail Naci, Cangül, One relator quotients of the modular group, Bull. Inst. Math. Acad. Sinica 32 (2004), no. 4, 291–296.Google Scholar
[41] J., Wiegold, The Schur multiplier: an elementary approach, Groups St Andrews 1981, London Math. Soc. Lecture Note Ser. 71 (Cambridge University Press, 1982), 137–154.Google Scholar
[42] Alun, Williams, Monoid Automata Factory, (2009), http://www.alunw.freeuk.com/MAF/maf.html

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