Pseudo-Anosov homeomorphisms and the lower central series of a surface group

Malestein, Justin¹

¹Department of Mathematics, University of Chicago, 5734 S University Ave, Chicago IL 60637, USA

Algebraic and Geometric Topology, Tome 7 (2007) no. 4, pp. 1921-1948

Résumé

Let Γk be the lower central series of a surface group Γ of a compact surface S with one boundary component. A simple question to ponder is whether a mapping class of S can be determined to be pseudo-Anosov given only the data of its action on Γ∕Γk for some k. In this paper, to each mapping class f which acts trivially on Γ∕Γk+1, we associate an invariant Ψk(f) ∈ End(H1(S, ℤ)) which is constructed from its action on Γ∕Γk+2 . We show that if the characteristic polynomial of Ψk(f) is irreducible over ℤ, then f must be pseudo-Anosov. Some explicit mapping classes are then shown to be pseudo-Anosov.

Détail

DOI : 10.2140/agt.2007.7.1921

Keywords: pseudo-Anosov, lower central series, Torelli group, Johnson filtration

Bibliographie

[1] H Bass, A Lubotzky, Linear-central filtrations on groups, from: "The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992)", Contemp. Math. 169, Amer. Math. Soc. (1994) 45

[2] M Bestvina, M Handel, Train-tracks for surface homeomorphisms, Topology 34 (1995) 109

[3] P Brinkmann, XTrain

[4] N Broaddus, B Farb, A Putman, Irreducible Sp–representations and subgroup distortion in the mapping class group, preprint

[5] A J Casson, S A Bleiler, Automorphisms of surfaces after Nielsen and Thurston, London Math. Soc. Student Texts 9, Cambridge University Press (1988)

[6] B Farb, C Leininger, D Margalit, The lower central series and pseudo-Anosov dilatations

[7] B Farb, D Margalit, A primer on mapping class groups, in preparation

[8] A Fathi, F Laudenbach, V Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France (1979) 284

[9] N V Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs 115, Amer. Math. Soc. (1992)

[10] D Johnson, An abelian quotient of the mapping class group $\cal{I}_{g}$, Math. Ann. 249 (1980) 225

[11] W Magnus, A Karrass, D Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers (John Wiley Sons) (1966)

[12] S Morita, On the structure of the Torelli group and the Casson invariant, Topology 30 (1991) 603

[13] S Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993) 699

[14] S Morita, The extension of Johnson's homomorphism from the Torelli group to the mapping class group, Invent. Math. 111 (1993) 197

[15] S Morita, Structure of the mapping class groups of surfaces: a survey and a prospect, from: "Proceedings of the Kirbyfest (Berkeley, CA, 1998)" (editors J Hass, M Scharlemann), Geom. Topol. Monogr. 2 (1999) 349

[16] R C Penner, A construction of pseudo-Anosov homeomorphisms, Trans. Amer. Math. Soc. 310 (1988) 179

[17] C Reutenauer, Free Lie algebras, London Math. Soc. Monographs, New Series 7, Oxford University Press (1993)

[18] W P Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. $($N.S.$)$ 19 (1988) 417