Geometric and Functional Analysis

, Volume 22, Issue 6, pp 1541–1590 | Cite as

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

  • Thomas KoberdaEmail author


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 .

Keywords and phrases

Subgroups of mapping class groups right-angled Artin groups 

Mathematics Subject Classification (1991)

Primary 37E30 Secondary 20F36 05C60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ago08.
    Ian Agol: Criteria for virtual fibering. Journal of Topology 1, 269–284 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. BR84.
    Baumslag G., Roseblade J.E.: Subgroups of direct products of free groups. Journal of the London Mathematical Society 30, 44–52 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  3. BC12.
    Jason Behrstock and Ruth Charney. Divergence and quasi–morphisms of right-angled Artin groups (2012). Preprint.Google Scholar
  4. BP92.
    Ricardo Benedetti and Carlo Petronio. Lectures on hyperbolic geometry. Universitext, Springer, Berlin (1992).Google Scholar
  5. BB97.
    Mladen Bestvina, Noel Brady: Morse theory and finiteness properties of groups. Inventiones Mathematicae 129, 123–139 (1997)Google Scholar
  6. BLM83.
    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.Google Scholar
  7. Bro12.
    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).Google Scholar
  8. Bri09.
    Martin R. Bridson. On the dimension of CAT(0) spaces where mapping class groups act. Preprint, 2009.Google Scholar
  9. BM12.
    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)Google Scholar
  10. Bro82.
    Kenneth S. Brown. Cohomology of groups. Graduate Texts in Mathematics, Vol. 87, Springer, New York (1982).Google Scholar
  11. Cha07.
    Ruth Charney: An introduction to right-angled Artin groups. Geometriae Dedicata 125, 141–158 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. CD95.
    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.Google Scholar
  13. CF12.
    Ruth Charney and Michael Farber. Random groups arising as graph products (2012). Preprint.Google Scholar
  14. CLM12.
    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)Google Scholar
  15. CF12.
    John Crisp and Benson Farb. The prevalence of surface groups in mapping class groups (2012). Preprint.Google Scholar
  16. CP01.
    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.Google Scholar
  17. CW07.
    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. Day09.
    Day Matthew B.: Peak reduction and finite presentations for automorphism groups of right-angled Artin groups. Geometry and Topology 13, 817–855 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Har00.
    Pierre de la Harpe. Topics in Geometric Group Theory. University of Chicago Press, (2000).Google Scholar
  20. Dro87a.
    Carl Droms: Subgroups of graph groups. Journal of Algebra 110, 519–522 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Dro87b.
    Carl Droms. Isomorphisms of graph groups. Proceedings of the American Mathematical Society (3)100 (1987), 407–408.Google Scholar
  22. Far06.
    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).Google Scholar
  23. FLM01.
    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. FM12.
    Benson Farb and Dan Margalit. A Primer on the mapping class group. Princeton Mathematical Series 49. Princeton University Press, Princeton, NJ (2012).Google Scholar
  25. FLP79.
    A. Fathi and F. Laudenbach and V. Poénaru. Travaux de Thurston sur les Surfaces. Soc. Math. de France, Paris, Astérisque 66-67 (1979).Google Scholar
  26. FP92.
    Edward Formanek and Claudio Procesi. The automorphism group of a free group is not linear. Journal of Algebra (2)149 (1992), 494–499.Google Scholar
  27. Fuj08.
    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.Google Scholar
  28. Fun12.
    Louis Funar. On power subgroups of mapping class groups (2012). (Preprint).Google Scholar
  29. Gol99.
    William M. Goldman. Complex hyperbolic geometry. Oxford Mathematical Monographs, 1999.Google Scholar
  30. Gub98.
    Joseph Gubeladze. The isomorphism problem for commutative monoid rings. Journal of Pure and Applied Algebra, (1)129 (1998), 35–65.Google Scholar
  31. HW08.
    Frédéric Haglund, Wise Daniel T.: Special cube complexes. GAFA, Geometric and Functional Analysis 17, 1551–1620 (2008)CrossRefzbMATHGoogle Scholar
  32. Ham06.
    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).Google Scholar
  33. Har86.
    John L. Harer. The virtual cohomological dimension of the mapping class group of an orientable surface. Inventiones mathematicae (1)84 (1986), 157–176.Google Scholar
  34. Iva92.
    Nikolai Ivanov. Subgroups of Teichmüller modular groups. Translations of Mathematical Monographs, 115 (1992).Google Scholar
  35. JS03.
    Tadeusz Januszkiewicz, Jacek Świa¸tkowski: Hyperbolic Coxeter groups of large dimension. Commentarii Mathematici Helvetici 78, 555–583 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  36. KL96.
    Michael Kapovich and Bernhard Leeb. Actions of discrete groups on nonpositively curved spaces. Mathematische Annalen (2)306 (1996), 341–352.Google Scholar
  37. Kat92.
    Svetlana Katok. Fuchsian groups. In: Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1992).Google Scholar
  38. KK12.
    Sang-hyun Kim and Thomas Koberda. Embedability between right-angled Artin groups. (2012) (Preprint).Google Scholar
  39. Kob12.
    Thomas Koberda. Geometry, Topology and Dynamics of Character Varieties. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, Vol. 23 (2012).Google Scholar
  40. Lau95.
    Michael R. Laurence. A generating set for the automorphism group of a graph group. Journal of London Mathematical Society (2)52 (1995), 318–334.Google Scholar
  41. Lev12.
    Gilbert Levitt. Unsolvability of the isomorphism problem for [free abelian]-by-free groups. (2012) (Preprint).Google Scholar
  42. Lon10.
    Michael Lönne. Presentations of subgroups of the braid group generated by powers of band generators. Topology and its Applications (7)157 (2010), 1127–1135.Google Scholar
  43. LR06.
    Chris Leininger, Alan Reid.: A combination theorem for Veech subgroups of the Mapping class group. Geometric and Functional Analysis 16, 403–436 (2006)MathSciNetCrossRefGoogle Scholar
  44. LS77.
    Roger C. Lyndon, Paul E. Schupp.: Combinatorial group theory. Springer, New York (1977)Google Scholar
  45. Mih68.
    K.A. Mihailova. The occurrence problem for free products of groups. (Russian) Mat. Sb. (N.S.) (117)75 (1968), 199–210.Google Scholar
  46. MP89.
    John McCarthy and Athanase Papadopoulos. Dynamics on Thurston’s sphere of projective measured foliations. Commentarii Mathematici Helvetici (1)64 (1989), 133-166.Google Scholar
  47. Mil71.
    Charles F. Miller, III. On group-theoretic decision problems and their classification. Annals of Mathematical Studies, 68 (1971).Google Scholar
  48. Mil92.
    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.Google Scholar
  49. Oh98.
    Hee Oh.: Discrete subgroups generated by lattices in opposite horospherical subgroups. Journal of Algebra 203, 621–676 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  50. PH92.
    R.C. Penner and J.L. Harer. Combinatorics of train tracks. In: Annals of Mathematics Studies, Vol. 125, Princeton University Press, Princeton (1992).Google Scholar
  51. Sab07.
    Lucas Sabalka. On rigidity and the isomorphism problem for tree braid groups. Groups, Geometry, and Dynamics, (3)3 (2009), 469-523.Google Scholar
  52. Sab09.
    Lucas Sabalka. On rigidity and the isomorphism problem for tree braid groups. Groups, Geometry, and Dynamics, (3)3 (2009), 469–523.Google Scholar
  53. Ser89.
    H. Servatius. Automorphisms of graph groups. Journal of Algebra (1)126 (1989), 34-60.Google Scholar
  54. Sta83.
    John Stallings. The topology of finite graphs. Inventiones mathematicae (3)71 (1983), 551-565.Google Scholar
  55. Sti87.
    John Stillwell. The occurrence problem for mapping class groups. Proceedings of the American Mathematical Society (3)101 (1987), 411-416.Google Scholar
  56. Tit72.
    Jacques Tits.: Free subgroups in linear groups. Journal of Algebra 20, 250–270 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  57. Wan07.
    Stephen Wang.: Representations of surface groups and right-angled Artin groups in higher rank. Algebraic and Geometric Topology 7, 1099–1117 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsYale UniversityNew HavenUSA

Personalised recommendations