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NIELSEN EQUIVALENCE OF GENERATING PAIRS OF SL(2,q)

Published online by Cambridge University Press:  25 February 2013

DARRYL MCCULLOUGH  and
MARCUS WANDERLEY
DARRYL MCCULLOUGH
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA e-mail: [email protected]
MARCUS WANDERLEY
Affiliation:
Departmento de Matematica, Universidade Federal de Pernambuco, Av. Prof. Luiz Freire, s/n Cid. Universitaria-Recife-PE, CEP 50.740-540, Brazil e-mail: [email protected]
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Abstract

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We present several conjectures which would describe the Nielsen equivalence classes of generating pairs for the groups SL(2,q) and PSL(2,q). The Higman invariant, which is the union of the conjugacy classes of the commutator of a generating pair and its inverse, and the trace of the commutator play key roles. Combining known results with additional work, we clarify the relationships between the conjectures, and obtain various partial results concerning them. Motivated by the work of Macbeath (A. M. Macbeath, Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, TX, 1967) (American Mathematical Society, Providence, RI, 1969), 14–32), we use another invariant defined using traces to develop algorithms that enable us to verify the conjectures computationally for all q up to 101, and to prove the conjectures for a highly restricted but possibly infinite set of q.

Keywords

Primary 20F99Secondary 20G40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 3 , September 2013 , pp. 481 - 509
Copyright
Copyright © Glasgow Mathematical Journal Trust 2013 

References

REFERENCES

1.Cohn, H., Approach to Markov's minimal forms through modular functions, Ann. Math. 61 (2) (1955), 1–12.Google Scholar
2.Collins, D. J., Generation and presentation of one-relator groups with centre, Math. Z. 157 (1977), 63–77.CrossRefGoogle Scholar
3.Dickson, L. E., Linear groups with an exposition of the Galois field theory (B.G. Tuebner, Leipzig, Germany, 1901; reprinted by Dover Publications, New York, 1958).Google Scholar
4.Dieudonné, J., On the automorphisms of the classical groups, with a supplement by Loo-Keng Hua, Mem. Amer. Math. Soc. 2 (1951), 1–122.Google Scholar
5.Dunwoody, M., On T-systems of groups, J. Aust. Math. Soc. 3 (1963), 172–179.CrossRefGoogle Scholar
6.Dunwoody, M., Nielsen transformations, in Computational problems in abstract algebra (Leech, J., Editor) (Proc. Conf. Oxford, 1967) (Pergamon, Oxford, UK, 1970), 45–46. MR 0260852 (41 #5472).Google Scholar
7.Evans, M. J., T-systems of certain finite simple groups, Math. Proc. Cambridge Phil. Soc. 113 (1) (1993), 9–22.Google Scholar
8.Gilman, R., Finite quotients of the automorphism group of a free group, Can. J. Math. 29 (1977), 541–551.Google Scholar
9.Glover, H. and Sjerve, D., The genus of PSL2(q), J. Reine Angew. Math. 380 (1987), 59–86.Google Scholar
10.GAP – Groups, Algorithms, and Programming. Available at http://www.gap-system.org/, accessed 4 January 2013.Google Scholar
11.Goldman, W., The modular group action on real SL(2)-characters of a one-holed torus, Geom. Topol. 7 (2003), 443–486.Google Scholar
12.Hua, L.-K., On the automorphisms of the symplectic group over any field, Ann. of Math. 49 (2) (1948), 739–759.Google Scholar
13.Lidl, R. and Niederreiter, H., Introduction to finite fields and their applications (Cambridge University Press, Cambridge, UK, 1986).Google Scholar
14.Lustig, M., Nielsen equivalence and simple-homotopy type, Proc. Lond. Math. Soc. 62 (3) (1991), 537–562.Google Scholar
15.Lustig, M. and Moriah, Y., Nielsen equivalence in Fuchsian groups and Seifert fibered spaces, Topology 30 (1991), 191–204.Google Scholar
16.Lustig, M. and Moriah, Y., Generalized Montesinos knots, tunnels and N-torsion, Math. Ann. 295 (1993), 167–189.Google Scholar
17.Lustig, M. and Moriah, Y., Generating systems of groups and Reidemeister–Whitehead torsion, J. Algebra 157 (1993), 170–198.CrossRefGoogle Scholar
18.Lustig, M. and Moriah, Y., N-torsion and applications, in Geometric group theory, vol. 1 (Proc. of Geometric Group Theory, University of Sussex, 1991), London Mathematical Society Lecture Note Series 181 (Cambridge University Press, Cambridge, UK, 1993), 159–168.Google Scholar
19.Lustig, M. and Moriah, Y., On the complexity of the Heegaard structure of hyperbolic 3-manifolds, Math. Z. 226 (1997), 349–358.CrossRefGoogle Scholar
20.Macbeath, A. M., Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, Tex., 1967) (American Mathematical Society, Providence, RI, 1969), 14–32.Google Scholar
21.McCullough, D., Exceptional subgroups of SL(2,F). Preprint available at www.math.ou.edu/~dmccullough/research/manuscripts.html, accessed 4 January 2013.Google Scholar
22.McCullough, D., Software for Nielsen equivalence of generating pairs of SL(2,q), GAP script. Available at www.math.ou.edu/~dmccullough/research/software.html, accessed 4 January 2013.CrossRefGoogle Scholar
23.McCullough, D. and Wanderley, M., Free actions on handlebodies, J. Pure Appl. Algebra 181 (2003), 85–104.Google Scholar
24.McCullough, D. and Wanderley, M., Writing elements of PSL(2,q) as commutators, Comm. Algebra 39 (2011), 1234–1241.Google Scholar
25.Moriah, Y., Heegaard splittings of Seifert fibered spaces, Invent. Math. 91 (1988), 465–481.Google Scholar
26.Neumann, B. H., On a question of Gaschütz, Arch. Math. (Basel) 7 (1956), 87–90.CrossRefGoogle Scholar
27.Neumann, B. H. and Neumann, H., Zwei Klassencharakteristischer Untergruppen und ihre Factorgruppen, Math. Nachr. 4 (1951) 106–125.Google Scholar
28.Nielsen, J., Die Isomorphismengruppe der allgemeinen unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 78 (1918), 385–397.CrossRefGoogle Scholar
29.Pride, S. J., The isomorphism problem for two-generator one-relator groups with torsion is solvable, Trans. Amer. Math. Soc. 227 (1977), 109–139.CrossRefGoogle Scholar
30.Rosenberger, G., All generating pairs of all two-generator Fuchsian groups, Arch. Math. 46 (1986), 198–204.Google Scholar
31.Schrier, O. and van der Waerden, B. L., Die Automorphismen der projektiven Gruppen, Abh. Math. Sem. Univ. Hamburg 6 (1928), 303–322.Google Scholar
32.Suzuki, M., Group theory, vol. I (Springer-Verlag, Berlin, Germany, 1982).Google Scholar
33.Webb, P. J., Minimal relation modules of free nilpotent groups, Arch. Math. (Basel) 37 (1981), 193–197.Google Scholar