Advertisement

Geometriae Dedicata

, Volume 175, Issue 1, pp 267–280 | Cite as

Relative second bounded cohomology of free groups

  • Cristina Pagliantini
  • Pascal Rolli
Original Paper
  • 95 Downloads

Abstract

This paper is devoted to the computation of the space \(\mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})\), where \(\Gamma \) is a free group of finite rank \(n\ge 2\) and \(H\) is a subgroup of finite rank. More precisely we prove that \(H\) has infinite index in \(\Gamma \) if and only if \(\mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})\) is not trivial, and furthermore, if and only if there is an isometric embedding \(\oplus _\infty ^n\mathcal {D}({\mathbb {Z}})\hookrightarrow \mathrm{H}_\mathrm{b}^2(\Gamma ,H;{\mathbb {R}})\), where \(\mathcal {D}({\mathbb {Z}})\) is the space of bounded alternating functions on \({\mathbb {Z}}\) equipped with the defect norm.

Keywords

Relative bounded cohomology Quasimorphisms Free groups  Split quasimorphisms Schreier graphs 

Mathematics Subject Classification

20J06 20E05 55N10 57M15 

Notes

Acknowledgments

We would like to thank Alessandro Sisto for pointing out how the results in [9] imply a weak version of our main theorem. Both authors were supported by Swiss National Science Foundation Project 144373, moreover the second author received support by Project 127016.

References

  1. 1.
    Bowditch, B.H.: Relatively hyperbolic groups. Int. J. Algebra Comput. 22(3), 1250016, 66 (2012)Google Scholar
  2. 2.
    Brooks, R.: Some remarks on bounded cohomology. In: Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference. State Univ. New York, Stony Brook, NY (1978); Ann. Math. Stud. vol. 97, pp. 53–63. Princeton Univ. Press, Princeton, NJ (1981)Google Scholar
  3. 3.
    Bestvina, M., Bromberg, K., Fujiwara, K.: Bounded Cohomology Via Quasi-trees. arXiv:1306.1542 (2013)
  4. 4.
    Burger, M., Monod, N.: Bounded cohomology of lattices in higher rank Lie groups. J. Eur. Math. Soc. 1, 199–235 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Burger, M., Monod, N.: Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal. 12, 219–280 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Calegari, D.: scl, Mathematical Society of Japan, (2009)Google Scholar
  7. 7.
    Dahmani, F., Guirardel, V., Osin, D.: Hyperbolically Embedded Subgroups and Rotating Families in Groups Acting on Hyperbolic Spaces. arXiv:1111.7048 (2011)
  8. 8.
    Frigerio, R., Pagliantini, C.: Relative measure homology and continuous bounded cohomology of topological pairs. Pac. J. Math. 257(1), 91–130 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Frigerio, R., Pozzetti, B., Sisto, A.: Extending Higher Dimensional Quasi-Cocycles. arXiv:1311.7633 (2013)
  10. 10.
    Gromov, M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. (1982), (56), 5–99 (1983)Google Scholar
  11. 11.
    Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) “Essay in Group Theory” Math. Sci. Res. Inst. Publ., Springer, New York, no. 8, pp. 75–263 (1987)Google Scholar
  12. 12.
    Hull, M., Osin, D.: Induced quasicocycles on groups with hyperbolically embedded subgroups. Algebr. Geom. Topol. (13), 2635–2665 (2013)Google Scholar
  13. 13.
    Ivanov, N.V.: Foundations of the theory of bounded cohomology, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), vol. 143, pp. 69–109, pp. 177–178, Studies in Topology, V (1985)Google Scholar
  14. 14.
    Ivanov, N.V.: Second bounded cohomology group. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167, 117–120 (1988)zbMATHGoogle Scholar
  15. 15.
    Löh, C.: Simplicial volume. Bull. Man. Atl., pp. 7–18, (2011). http://www.map.mpim-bonn.mpg.de/Simplicial_volume
  16. 16.
    Matsumoto, S., Morita, S.: Bounded cohomology of certain groups of homeomorphisms. Proc. Am. Math. Soc. 94, 539–544 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Mineyev, I.: Straightening and bounded cohomology of hyperbolic groups. GAFA Geom. Funct. Anal. 11, 807–839 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Mineyev, I.: Bounded cohomology characterizes hyperbolic groups. Q. J. Math. Oxf. Ser. 53, 59–73 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Mitsumatsu, Y.: Bounded cohomology and \(l^1\)-homology of surfaces. Topology 23, 465–471 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Monod, N.: Continuous Bounded Cohomology of Locally Compact Groups, Lecture Notes in Mathematics, vol. 1758. Springer, Berlin (2001)Google Scholar
  21. 21.
    Pagliantini, C.: Relative (Continuous) Bounded Cohomology and Simplicial Volume of Hyperbolic Manifolds with Geodesic Boundary. PhD thesis, Università di Pisa (2012). http://etd.adm.unipi.it/theses/available/etd-07112012-101103/
  22. 22.
    Park, H.S.: Relative bounded cohomology. Topol. Appl. 131, 203–234 (2003)CrossRefzbMATHGoogle Scholar
  23. 23.
    Rolli, P.: Quasi-Morphisms on Free Groups. arXiv:0911.4234 (2009)
  24. 24.
    Rolli, P.: Split Quasicocycles. arXiv:1305.0095 (2013)
  25. 25.
    Rolli, P.: Split Quasicocycles and Defect Spaces. PhD thesis, ETH Zurich (2014). doi: 10.3929/ethz-a-010168196
  26. 26.
    Stallings, J.R.: Topology of finite graphs. Invent. Math. 71, 551–565 (1983)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department MathematikETH ZürichZurichSwitzerland

Personalised recommendations