Skip to main content
SpringerLink
Log in

Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

Consider the mapping class group Mod g,p of a surface Σ g,p of genus g with p punctures, and a finite collection {f1, . . . , fk} of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large N, the mapping classes \({\{f_1^N,\ldots,f_k^N\}}\) generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When {f1, . . . , fk} are arbitrary mapping classes, we show that sufficiently large powers of these mapping classes generate a group which embeds in a right-angled Artin group in a controlled way. We establish some analogous results for real and complex hyperbolic manifolds. We also discuss the unsolvability of the isomorphism problem for finitely generated subgroups of Mod g,p , and recover the fact that the isomorphism problem for right-angled Artin groups is solvable. We thus characterize the isomorphism type of many naturally occurring subgroups of Mod g,p .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ian Agol: Criteria for virtual fibering. Journal of Topology 1, 269–284 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baumslag G., Roseblade J.E.: Subgroups of direct products of free groups. Journal of the London Mathematical Society 30, 44–52 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Jason Behrstock and Ruth Charney. Divergence and quasi–morphisms of right-angled Artin groups (2012). Preprint.

  4. Ricardo Benedetti and Carlo Petronio. Lectures on hyperbolic geometry. Universitext, Springer, Berlin (1992).

  5. Mladen Bestvina, Noel Brady: Morse theory and finiteness properties of groups. Inventiones Mathematicae 129, 123–139 (1997)

    Google Scholar 

  6. Joan S. Birman and Alex Lubotzky and John McCarthy. Abelian and solvable subgroups of the mapping class groups. Duke Mathematical Journal (4)50 (1983), 1107–1120.

  7. Martin R. Bridson. Semisimple actions of mapping class groups on CAT(0) spaces. The Geometry of Riemann Surfaces, LMS Lecture Notes, Vol. 368 (2012) (To appear).

  8. Martin R. Bridson. On the dimension of CAT(0) spaces where mapping class groups act. Preprint, 2009.

  9. Martin R. Bridson and Charles F. Miller III. Structure and finiteness properties of subdirect products of groups. Proceedings of London Mathematical Society (2012) (To appear)

  10. Kenneth S. Brown. Cohomology of groups. Graduate Texts in Mathematics, Vol. 87, Springer, New York (1982).

  11. Ruth Charney: An introduction to right-angled Artin groups. Geometriae Dedicata 125, 141–158 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ruth Charney and Michael W. Davis. Finite K(π, 1)’s for Artin groups. Prospects in topology (Princeton, NJ, 1994), In: Annals of Mathematical Studies, Vol. 138, Princeton Univ. Press, Princeton (1995), pp. 110–124.

  13. Ruth Charney and Michael Farber. Random groups arising as graph products (2012). Preprint.

  14. Matt Clay, Chris Leininger and Johanna Mangahas. The geometry of right angled Artin subgroups of mapping class groups. Groups, Geometry, and Dynamics (2012) (To appear)

  15. John Crisp and Benson Farb. The prevalence of surface groups in mapping class groups (2012). Preprint.

  16. John Crisp and Luis Paris. The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group. Inventiones Mathematicae (1)145 (2001), 19–36.

  17. John Crisp and Bert Wiest. Quasi-isometrically embedded subgroups of braid and diffeomorphism groups. Transactions of American Mathematical Society (11)359 (2007), 5485–5503.

    Google Scholar 

  18. Day Matthew B.: Peak reduction and finite presentations for automorphism groups of right-angled Artin groups. Geometry and Topology 13, 817–855 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pierre de la Harpe. Topics in Geometric Group Theory. University of Chicago Press, (2000).

  20. Carl Droms: Subgroups of graph groups. Journal of Algebra 110, 519–522 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. Carl Droms. Isomorphisms of graph groups. Proceedings of the American Mathematical Society (3)100 (1987), 407–408.

  22. Benson Farb. Some problems on mapping class groups and moduli space. In Problems on Mapping Class Groups and Related Topics. Proc. Symp. Pure and Applied Math., Vol. 74, (2006).

  23. Benson Farb and Alexander Lubotzky and Yair Minsky. Rank one phenomena for mapping class groups. Duke Mathematical Journal, Vol. 106, No. 3, 581–597 (2001).

    Google Scholar 

  24. Benson Farb and Dan Margalit. A Primer on the mapping class group. Princeton Mathematical Series 49. Princeton University Press, Princeton, NJ (2012).

  25. A. Fathi and F. Laudenbach and V. Poénaru. Travaux de Thurston sur les Surfaces. Soc. Math. de France, Paris, Astérisque 66-67 (1979).

  26. Edward Formanek and Claudio Procesi. The automorphism group of a free group is not linear. Journal of Algebra (2)149 (1992), 494–499.

  27. Koji Fujiwara. Subgroups generated by two pseudo-Anosov elements in a mapping class group. I. Uniform exponential growth. Groups of diffeomorphisms, Adv. Stud. Pure Math., Vol. 52, Math. Soc. Japan, Tokyo (2008), pp. 283–296.

  28. Louis Funar. On power subgroups of mapping class groups (2012). (Preprint).

  29. William M. Goldman. Complex hyperbolic geometry. Oxford Mathematical Monographs, 1999.

  30. Joseph Gubeladze. The isomorphism problem for commutative monoid rings. Journal of Pure and Applied Algebra, (1)129 (1998), 35–65.

  31. Frédéric Haglund, Wise Daniel T.: Special cube complexes. GAFA, Geometric and Functional Analysis 17, 1551–1620 (2008)

    Article  MATH  Google Scholar 

  32. Ursula Hamenstädt. Geometric properties of the mapping class group. In Problems on Mapping Class Groups and Related Topics. Proc. Symp. Pure and Applied Math., Vol. 74 (2006).

  33. John L. Harer. The virtual cohomological dimension of the mapping class group of an orientable surface. Inventiones mathematicae (1)84 (1986), 157–176.

  34. Nikolai Ivanov. Subgroups of Teichmüller modular groups. Translations of Mathematical Monographs, 115 (1992).

  35. Tadeusz Januszkiewicz, Jacek Świa¸tkowski: Hyperbolic Coxeter groups of large dimension. Commentarii Mathematici Helvetici 78, 555–583 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Michael Kapovich and Bernhard Leeb. Actions of discrete groups on nonpositively curved spaces. Mathematische Annalen (2)306 (1996), 341–352.

  37. Svetlana Katok. Fuchsian groups. In: Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1992).

  38. Sang-hyun Kim and Thomas Koberda. Embedability between right-angled Artin groups. (2012) (Preprint).

  39. Thomas Koberda. Geometry, Topology and Dynamics of Character Varieties. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, Vol. 23 (2012).

  40. Michael R. Laurence. A generating set for the automorphism group of a graph group. Journal of London Mathematical Society (2)52 (1995), 318–334.

  41. Gilbert Levitt. Unsolvability of the isomorphism problem for [free abelian]-by-free groups. (2012) (Preprint).

  42. Michael Lönne. Presentations of subgroups of the braid group generated by powers of band generators. Topology and its Applications (7)157 (2010), 1127–1135.

  43. Chris Leininger, Alan Reid.: A combination theorem for Veech subgroups of the Mapping class group. Geometric and Functional Analysis 16, 403–436 (2006)

    Article  MathSciNet  Google Scholar 

  44. Roger C. Lyndon, Paul E. Schupp.: Combinatorial group theory. Springer, New York (1977)

    Google Scholar 

  45. K.A. Mihailova. The occurrence problem for free products of groups. (Russian) Mat. Sb. (N.S.) (117)75 (1968), 199–210.

  46. John McCarthy and Athanase Papadopoulos. Dynamics on Thurston’s sphere of projective measured foliations. Commentarii Mathematici Helvetici (1)64 (1989), 133-166.

  47. Charles F. Miller, III. On group-theoretic decision problems and their classification. Annals of Mathematical Studies, 68 (1971).

  48. Chales F. Miller III. Decision problems for groupssurvey and reflections, in Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., Vol. 23, Springer, Berlin (1992), pp. 1–59.

  49. Hee Oh.: Discrete subgroups generated by lattices in opposite horospherical subgroups. Journal of Algebra 203, 621–676 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  50. R.C. Penner and J.L. Harer. Combinatorics of train tracks. In: Annals of Mathematics Studies, Vol. 125, Princeton University Press, Princeton (1992).

  51. Lucas Sabalka. On rigidity and the isomorphism problem for tree braid groups. Groups, Geometry, and Dynamics, (3)3 (2009), 469-523.

  52. Lucas Sabalka. On rigidity and the isomorphism problem for tree braid groups. Groups, Geometry, and Dynamics, (3)3 (2009), 469–523.

  53. H. Servatius. Automorphisms of graph groups. Journal of Algebra (1)126 (1989), 34-60.

  54. John Stallings. The topology of finite graphs. Inventiones mathematicae (3)71 (1983), 551-565.

  55. John Stillwell. The occurrence problem for mapping class groups. Proceedings of the American Mathematical Society (3)101 (1987), 411-416.

  56. Jacques Tits.: Free subgroups in linear groups. Journal of Algebra 20, 250–270 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  57. Stephen Wang.: Representations of surface groups and right-angled Artin groups in higher rank. Algebraic and Geometric Topology 7, 1099–1117 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Yale University, 208283, New Haven, CT, 06511-8283, USA

    Thomas Koberda

Authors
  1. Thomas Koberda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Koberda.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Koberda, T. Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups. Geom. Funct. Anal. 22, 1541–1590 (2012). https://doi.org/10.1007/s00039-012-0198-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-012-0198-z

Keywords and phrases

Mathematics Subject Classification (1991)

Search

Navigation