A ``Tits-alternative'' for subgroups of surface mapping class groups

Author:
John McCarthy

Journal:
Trans. Amer. Math. Soc. **291** (1985), 583-612

MSC:
Primary 57M99; Secondary 20F38, 57N05

MathSciNet review:
800253

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Abstract: It has been observed that surface mapping class groups share various properties in common with the class of linear groups (e.g., ). In this paper, the known list of such properties is extended to the ``Tits-Alternative'', a property of linear groups established by J. Tits . In fact, we establish that every subgroup of a surface mapping class group is either virtually abelian or contains a nonabelian free group.

In addition, in order to establish this result, we develop a theory of attractors and repellers for the action of surface mapping classes on Thurston's projective lamination spaces . This theory generalizes results known for pseudo-Anosov mapping classes .

**[**Hyman Bass and Alexander Lubotzky,**BL**]*Automorphisms of groups and of schemes of finite type*, Israel J. Math.**44**(1983), no. 1, 1–22. MR**693651**, 10.1007/BF02763168**[**Joan S. Birman,**B1**]*Braids, links, and mapping class groups*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. MR**0375281****[**Joan S. Birman,**B2**]*The algebraic structure of surface mapping class groups*, Discrete groups and automorphic functions (Proc. Conf., Cambridge, 1975), Academic Press, London, 1977, pp. 163–198. MR**0488019****[**Joan S. Birman, Alex Lubotzky, and John McCarthy,**BLM**]*Abelian and solvable subgroups of the mapping class groups*, Duke Math. J.**50**(1983), no. 4, 1107–1120. MR**726319**, 10.1215/S0012-7094-83-05046-9**[****FLP**]*Travaux de Thurston sur les surfaces*, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR**568308****[**Jane Gilman,**G**]*On the Nielsen type and the classification for the mapping class group*, Adv. in Math.**40**(1981), no. 1, 68–96. MR**616161**, 10.1016/0001-8708(81)90033-5**[**John L. Harer,**H**]*Stability of the homology of the mapping class groups of orientable surfaces*, Ann. of Math. (2)**121**(1985), no. 2, 215–249. MR**786348**, 10.2307/1971172**[**R. C. Penner and J. L. Harer,**HP**]*Combinatorics of train tracks*, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. MR**1144770****[**W. J. Harvey,**Ha**]*Geometric structure of surface mapping class groups*, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 255–269. MR**564431****[**A. Hatcher and W. Thurston,**HT**]*A presentation for the mapping class group of a closed orientable surface*, Topology**19**(1980), no. 3, 221–237. MR**579573**, 10.1016/0040-9383(80)90009-9**[**Steven P. Kerckhoff,**Ke**]*The Nielsen realization problem*, Ann. of Math. (2)**117**(1983), no. 2, 235–265. MR**690845**, 10.2307/2007076**[**J. McCarthy,**Mc1**]*Normalizers and centralizers of pseudo-Anosov mapping classes*, preprint available upon request.**[**-,**Mc2**]*Subgroups of surface mapping class groups*, Ph.D. thesis, Columbia University, May, 1983.**[**J. Morgan,**M**]*Train tracks and geodesic laminations*, Columbia University Lecture Notes (to appear).**[**Jakob Nielsen,**N**]*Surface transformation classes of algebraically finite type*, Danske Vid. Selsk. Math.-Phys. Medd.**21**(1944), no. 2, 89. MR**0015791****[**R. Penner,**P**]*A computation of the action of the mapping class group on isotopy classes of curves and arcs in surfaces*, Ph.D. thesis, Massachusetts Institute of Technology, 1982.**[**W. P. Thurston,**Th1**]*The geometry and topology of**-manifolds*, Princeton Univ. Lecture Notes (to appear).**[**-,**Th2**]*On the geometry and dynamics of diffeomorphisms of surfaces*, preprint.**[**William P. Thurston,**Th3**]*Three-dimensional manifolds, Kleinian groups and hyperbolic geometry*, Bull. Amer. Math. Soc. (N.S.)**6**(1982), no. 3, 357–381. MR**648524**, 10.1090/S0273-0979-1982-15003-0**[**-, Lectures Notes, Boulder, Colorado, 1980.**Th4**]**[**-,**Th5**]*Hyperbolic structures on**-manifolds*. II, preprint, July, 1980.**[**J. Tits,**T**]*Free subgroups in linear groups*, J. Algebra**20**(1972), 250–270. MR**0286898**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0800253-8

Keywords:
Surface mapping class group,
geodesic lamination,
virtually abelian

Article copyright:
© Copyright 1985
American Mathematical Society