Glasnik Matematicki, Vol. 49, No. 2 (2014), 337-350.

ON FINITE INDEX SUBGROUPS OF THE MAPPING CLASS GROUP OF A NONORIENTABLE SURFACE

Błażej Szepietowski

Institute of Mathematics, Gdansk University, Wita Stwosza 57, 80-952 Gdansk, Poland
e-mail: blaszep@mat.ug.edu.pl


Abstract.   Let M(Nh,n) denote the mapping class group of a compact nonorientable surface of genus h ≥ 7 and n ≤ 1 boundary components, and let T(Nh,n) be the subgroup of M(Nh,n) generated by all Dehn twists. It is known that T(Nh,n) is the unique subgroup of M(Nh,n) of index 2. We prove that T(Nh,n) (and also M(Nh,n)) contains a unique subgroup of index 2g-1(2g-1) up to conjugation, and a unique subgroup of index 2g-1(2g+1) up to conjugation, where g = ⌊(h-1)/2⌋. The other proper subgroups of T(Nh,n) and M(Nh,n) have index greater than 2g-1(2g+1). In particular, the minimum index of a proper subgroup of T(Nh,n) is 2g-1(2g-1).

2010 Mathematics Subject Classification.   20F38, 57N05.

Key words and phrases.   Mapping class group, nonorinatble surface, finite index subgroup.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.49.2.08


References:

  1. J. A. Berrick, V. Gebhardt and L. Paris, Finite index subgroups of mapping class groups, Proc. London Math. Soc. (3) 108 (2014), 575-599.
    MathSciNet     CrossRef

  2. J. S. Birman and D. R. J. Chillingworth, On the homeotopy group of a non-orientable surface, Proc. Cambridge Philos. Soc. 71 (1972), 437-448.
    MathSciNet     CrossRef

  3. D. R. J. Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Proc. Cambridge Philos. Soc. 65 (1969), 409-430.
    MathSciNet     CrossRef

  4. E. K. Grossman, On the residual finiteness of certain mapping class groups, J. London Math. Soc. (2) 9 (1974/75) 160-164.
    MathSciNet     CrossRef

  5. S. P. Humphries, Generators for the mapping class group, in: Topology of low dimensional manifolds (ed. R. Fenn), Springer-Verlag, Berlin, 1979, 44-47.
    MathSciNet    

  6. M. Korkmaz, First homology group of mapping class group of nonorientable surfaces, Math. Proc. Cambridge Philos. Soc. 123 (1998), 487-499.
    MathSciNet     CrossRef

  7. M. Korkmaz, Low-dimensional homology groups of mapping class groups: a survey, Turkish J. Math. 26 (2002), 101-114.
    MathSciNet    

  8. W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc. 59 (1963), 307-317.
    MathSciNet     CrossRef

  9. W. B. R. Lickorish, On the homeomorphisms of a non-orientable surface, Proc. Cambridge Philos. Soc. 61 (1965), 61-64.
    MathSciNet     CrossRef

  10. G. Masbaum and A. W. Reid, All finite groups are involved in the mapping class group, Geom. Topol. 16 (2012), 1393-1411.
    MathSciNet     CrossRef

  11. J. D. McCarthy and U. Pinkall, Representing homology automorphisms of nonorientable surfaces, Max Planc Inst. preprint MPI/SFB 85-11, revised version written in 2004. Available at http://www.math.msu.edu/ mccarthy.

  12. L. Paris, Small index subgroups of the mapping class group, J. Group Theory 13 (2010), 613-618.
    MathSciNet     CrossRef

  13. L. Paris and B. Szepietowski, A presentation for the mapping class group of a nonorientable surface, preprint, arXiv:1308.5856.

  14. M. Stukow, Commensurability of geometric subgroups of mapping class groups, Geom. Dedicata 143 (2009), 117-142.
    MathSciNet     CrossRef

  15. M. Stukow, The twist subgroup of the mapping class group of a nonorientable surface, Osaka J. Math. 46 (2009), 717-738.
    MathSciNet     CrossRef

  16. M. Stukow, Generating mapping class groups of nonorientable surfaces with boundary, Adv. Geom. 10 (2010), 249-273.
    MathSciNet     CrossRef

  17. B. Szepietowski, Embedding the braid group in mapping class groups, Publ. Mat. 54 (2010), 359-368.
    MathSciNet     CrossRef

  18. B. Szepietowski, Low-dimensional linear representations of the mapping class group of a nonorientable surface, Algebr. Geom. Topol. 14 (2014), 565-594.
    MathSciNet     CrossRef

  19. B. Zimmermann, On minimal finite quotients of mapping class groups, Rocky Mountain J. Math. 42 (2012), 1411-1420.
    MathSciNet     CrossRef

Glasnik Matematicki Home Page