Finite abelian subgroups of the mapping class group

Broughton, Allen¹ ; Wootton, Aaron²

¹Department of Mathematics, Rose Hulman Institute of Technology, Terre Haute, IN 47803, USA
²Department of Mathematics, The University of Portland, Portland, OR 97203, USA

Algebraic and Geometric Topology, Tome 7 (2007) no. 4, pp. 1651-1697

Résumé

The problem of enumeration of conjugacy classes of finite abelian subgroups of the mapping class group ℳσ of a closed, smooth, orientable surface S of genus σ ≥ 2 is considered. A complete method of enumeration is achieved for finite elementary abelian subgroups and steps are taken toward enumeration of finite abelian subgroups.

Détail

DOI : 10.2140/agt.2007.7.1651

Keywords: finite subgroups of mapping class groups, automorphism groups of surfaces

Bibliographie

[1] I Averill, J Gregoire, Tilings of Low-Genus Surfaces by Quadrilaterals (2002)

[2] L Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978) 73

[3] J S Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies 82, Princeton University Press (1974)

[4] T E Brendle, The Torelli Group and Representations of Mapping Class Groups, PhD thesis, Columbia University (2002)

[5] T E Brendle, B Farb, Every mapping class group is generated by 6 involutions, J. Algebra 278 (2004) 187

[6] T Breuer, Characters and automorphism groups of compact Riemann surfaces, London Mathematical Society Lecture Note Series 280, Cambridge University Press (2000)

[7] S A Broughton, The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups, Topology Appl. 37 (1990) 101

[8] S A Broughton, Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69 (1991) 233

[9] S A Broughton, Normalizers and centralizers of elementary abelian subgroups of the mapping class group, from: "Topology '90 (Columbus, OH, 1990)", Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 77

[10] S A Broughton, Enumeration of the Equisymmetric Strata of the Moduli Space of Surfaces of Low Genus (2005)

[11] S A Broughton, R Dirks, M Sloughter, C Vinroot, Triangular Surface Tiling Groups for Genus 2 to 13 (2002)

[12] J Cannon, W Bosma, editors, Handbook of Magma Functions, Edition 2.13 (2006) 4350

[13] M D E Conder, R S Kulkarni, Infinite families of automorphism groups of Riemann surfaces, from: "Discrete groups and geometry (Birmingham, 1991)", London Math. Soc. Lecture Note Ser. 173, Cambridge Univ. Press (1992) 47

[14] G Group, GAP – Groups, Algorithms and Programming, Version 4.4 (2006)

[15] J Gilman, Prime Order Automorphisms of Riemann Surfaces

[16] J Gilman, On conjugacy classes in the Teichmüller modular group, Michigan Math. J. 23 (1976) 53

[17] J Gilman, A matrix representation for automorphisms of compact Riemann surfaces, Linear Algebra and Appl. 17 (1977) 139

[18] J Gilman, Structures of elliptic irreducible subgroups of the modular group, Proc. London Math. Soc. $(3)$ 47 (1983) 27

[19] J Gilman, D Patterson, Intersection matrices for bases adapted to automorphisms of a compact Riemann surface, from: "Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978)", Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 149

[20] W J Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math. Oxford Ser. $(2)$ 17 (1966) 86

[21] W J Harvey, On branch loci in Teichmüller space, Trans. Amer. Math. Soc. 153 (1971) 387

[22] D Johnson, A survey of the Torelli group, from: "Low-dimensional topology (San Francisco, CA, 1981)", Contemp. Math. 20, Amer. Math. Soc. (1983) 165

[23] S P Kerckhoff, The Nielsen realization problem, Ann. of Math. $(2)$ 117 (1983) 235

[24] R S Kulkarni, Infinite families of surface symmetries, Israel J. Math. 76 (1991) 337

[25] A M Macbeath, The classification of non-euclidean plane crystallographic groups, Canad. J. Math. 19 (1967) 1192

[26] A M Macbeath, Generators of the linear fractional groups, from: "Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967)", Amer. Math. Soc. (1969) 14

[27] C Maclachlan, Abelian groups of automorphisms of compact Riemann surfaces, Proc. London Math. Soc. $(3)$ 15 (1965) 699

[28] C Maclachlan, Modulus space is simply-connected, Proc. Amer. Math. Soc. 29 (1971) 85

[29] Maplesoft, Maple, Version 10

[30] D Singerman, Subgroups of Fuschian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970) 319

[31] D Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. $(2)$ 6 (1972) 29

[32] D Singerman, Symmetries of Riemann surfaces with large automorphism group, Math. Ann. 210 (1974) 17

[33] D Singerman, Automorphisms of maps, permutation groups and Riemann surfaces, Bull. London Math. Soc. 8 (1976) 65

[34] A Wootton, Non-normal Belyĭ $p$-gonal surfaces, from: "Computational aspects of algebraic curves" (editor T Shaska), Lecture Notes Ser. Comput. 13, World Sci. Publ., Hackensack, NJ (2005) 95

[35] A Wootton, The full automorphism group of a cyclic $p$-gonal surface, J. Algebra 312 (2007) 377

[36] H Zieschang, E Vogt, H D Coldewey, Surfaces and planar discontinuous groups, Lecture Notes in Mathematics 835, Springer (1980)