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Marked fatgraph complexes and surface automorphisms

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Abstract

Combinatorial aspects of the Torelli–Johnson–Morita theory of surface automorphisms are extended to certain subgroups of the mapping class groups. These subgroups are defined relative to a specified homomorphism from the fundamental group of the surface onto an arbitrary group K. For K abelian, there is a combinatorial theory akin to the classical case, for example, providing an explicit cocycle representing the first Johnson homomophism with target Λ 3 K. Furthermore, the Earle class with coefficients in K is represented by an explicit cocyle.

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References

  1. Andersen J., Bene A., Penner R.C.: Groupoid lifts of mapping class representations for bordered surfaces. Top. Appl. 156, 2713–2735 (2010)

    Article  MathSciNet  Google Scholar 

  2. Andersen J.E., Bene A., Meilhan J.B., Penner R.C.: Finite type invariants and fatgraphs. Adv. Math. 225, 2117–2161 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bene A., Kawazumi N., Penner R.C.: Canonical extensions of the Johnson homomorphisms to the Torelli groupoid. Adv. Math. 221, 627–659 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Broaddus N., Farb B., Putman A.: Irreducible Sp-representations and subgroup distortion in the mapping class group. Comment. Math. Helv. 86, 537–556 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Earle C.J.: Families of Riemann surfaces and Jacobi varieties. Ann. Math. 107, 255–286 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Johnson D.: An abelian quotient of the mapping class group \({\fancyscript{I}_g}\). Math. Ann. 249, 225–242 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. Johnson D.: A survey of the Torelli group. Contemp. Math. 20, 165–179 (1983)

    Article  MATH  Google Scholar 

  8. Johnson D.: The structure of the Torelli group, III. The abelianization of \({\fancyscript{I}_g}\). Topology 24(2), 113–126 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kawazumi, N.: Cohomological aspects of Magnus expansions, preprint, math.GT/0505497 (2005)

  10. Kawazumi N.: Canonical 2-forms on the moduli of Riemann surfaces. In: Papadopoulos, A. (ed.) Handbook of Teichmüller theory, Vol. II, pp. 217–237. EMS Publishing House, Zurich (2009)

    Chapter  Google Scholar 

  11. Kuno Y.: A combinatorial formula for Earle’s twisted 1-cocycle on the mapping class group M g,*. Math. Proc. Camb. Phil. Soc. 146, 109–118 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Labute J.: On the descending central series of groups with a single defining relation. J. Algebra 14, 16–23 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  13. Magnus W., Karrass A., Solitar D.: Combinatorial group theory. Dover Publications, New York (2004)

    MATH  Google Scholar 

  14. Massuyeau, G.: Canonical extensions of Morita homomorphisms to the Ptolemy groupoid. Geom. Dedicata, to appear, doi:10.1007/s10711-011-9640-x; arXiv:1006.0920v3 [math.GT]

  15. McMillan D.R.: Homeomorphisms on a solid torus. Proc. Am. Math. Soc. 14, 386–390 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  16. Morita S.: Families of Jacobian manifolds and characteristic classes of surface bundles I. Ann. Inst. Fourier 39, 777–810 (1989)

    Article  MATH  Google Scholar 

  17. Morita S.: Abelian quotients of subgroups of the mapping class group of surfaces. Duke Math. J. 70, 699–726 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Morita S.: Structure of the mapping class groups of surfaces: a survey and a prospect. Geometry and Topology Monographs Vol. 2. Proceedings of the Kirbyfest, pp. 349–406 (1999)

  19. Morita S., Penner R.C.: Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves. Math. Proc. Camb. Phil. Soc. 144, 651–671 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Penner, R.C.: Decorated Teichmüller Theory, QGM Masters Class Series, European Mathematical Society (2011)

  21. Perron B.: Filtration de Johnson et groupe de Torelli modulo p ,p premier. C. R. Math. Acad. Sci. Paris 346(11-12), 667–670 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Raghunathan M.S.: Cohomology of arithmetic subgroups of algebraic groups: I. Ann. Math. 86, 409–424 (1967)

    Article  MATH  Google Scholar 

  23. Sato M.: The abelianization of the level d mapping class group. J. Topol 3(4), 847–882 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Suzuki S.: On homeomorphisms of a 3-dimensional handlebody. Can. J. Math. 29, 111–124 (1977)

    Article  MATH  Google Scholar 

  25. Zieschang, H., Vogt, E., Coldeway, H.D.: Surfaces and planar discontinuous groups. Lecture Notes in Mathematics, Vol. 835, (1980)

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Authors and Affiliations

  1. Department of Mathematics, Tsuda College, 2-1-1 Tsuda-Machi, Kodaira-shi, Tokyo, 187-8577, Japan

    Yusuke Kuno

  2. Center for the Quantum Geometry of Moduli Spaces, Aarhus University, 8000, C Aarhus, Denmark

    R. C. Penner

  3. Departments of Mathematics and Theoretical Physics, Caltech, Pasadena, CA, 91125, USA

    R. C. Penner

  4. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Vladimir Turaev

Authors
  1. Yusuke Kuno
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  2. R. C. Penner
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  3. Vladimir Turaev
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Correspondence to Yusuke Kuno.

Additional information

This work was done at the Center for Quantum Geometry of Moduli Spaces funded by the Danish National Research Foundation.

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Kuno, Y., Penner, R.C. & Turaev, V. Marked fatgraph complexes and surface automorphisms. Geom Dedicata 167, 151–166 (2013). https://doi.org/10.1007/s10711-012-9807-0

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