On the linearity problem for mapping class groups

Brendle, Tara E¹ ; Hamidi-Tehrani, Hessam²

¹Columbia University, Department of Mathematics, New York NY 10027, USA
²B.C.C. of the City University of New York, Department of Mathematics and Computer Science, Bronx NY 10453, USA

Algebraic and Geometric Topology, Tome 1 (2001) no. 1, pp. 445-468

Résumé

Formanek and Procesi have demonstrated that Aut(Fn) is not linear for n ≥ 3. Their technique is to construct nonlinear groups of a special form, which we call FP-groups, and then to embed a special type of automorphism group, which we call a poison group, in Aut(Fn), from which they build an FP-group. We first prove that poison groups cannot be embedded in certain mapping class groups. We then show that no FP-groups of any form can be embedded in mapping class groups. Thus the methods of Formanek and Procesi fail in the case of mapping class groups, providing strong evidence that mapping class groups may in fact be linear.

Détail

DOI : 10.2140/agt.2001.1.445

Keywords: mapping class group, linearity, poison group

Bibliographie

[1] H Bass, Covering theory for graphs of groups, J. Pure Appl. Algebra 89 (1993) 3

[2] S J Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001)

[3] S J Bigelow, R D Budney, The mapping class group of a genus two surface is linear, Algebr. Geom. Topol. 1 (2001)

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

[5] J S Birman, A Lubotzky, J Mccarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983) 1107

[6] K S Brown, Cohomology of groups, Graduate Texts in Mathematics, Springer-Verlag (1982)

[7] J L Dyer, E Formanek, E K Grossman, On the linearity of automorphism groups of free groups, Arch. Math. (Basel) 38 (1982) 404

[8] D B A Epstein, Curves on 2–manifolds and isotopies, Acta Math. 115 (1966) 83

[9] E Formanek, C Procesi, The automorphism group of a free group is not linear, J. Algebra 149 (1992) 494

[10] J Hass, P Scott, Shortening curves on surfaces, Topology 33 (1994) 25

[11] N V Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, American Mathematical Society (1992)

[12] G Levitt, K Vogtmann, A Whitehead algorithm for surface groups, Topology 39 (2000) 1239

[13] M Korkmaz, On the linearity of certain mapping class groups, Turkish J. Math. 24 (2000) 367

[14] D Krammer, The braid group $B_4$ is linear, Invent. Math. 142 (2000) 451

[15] W Magnus, A Karrass, D Solitar, Combinatorial group theory, Dover Publications Inc. (1976)

[16] J Mccarthy, A “Tits-alternative” for subgroups of surface mapping class groups, Trans. Amer. Math. Soc. 291 (1985) 583

[17] J Nielsen, Die Gruppe der dreidimensionalen Gittertransformationen, Danske Vid. Selsk. Mat-Fys. (1924) 1

[18] D Rolfsen, Knots and links, Mathematics Lecture Series, Publish or Perish Inc. (1990)

[19] B A F Wehrfritz, Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices, Ergebnisse der Matematik und ihrer Grenzgebiete 76, Springer-Verlag (1973)