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Relative bounded cohomology for groupoids

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Abstract

We introduce bounded cohomology for (pairs of) groupoids and develop homological algebra to deal with it. We generalise results of Ivanov, Frigerio and Pagliantini to this setting and show that (under topological conditions) the bounded cohomology of a pair of topological spaces is isometrically isomorphic to the bounded cohomology of the pair of fundamental groupoids. Furthermore, we prove that bounded cohomology relative to an amenable groupoid is isometrically isomorphic to the bounded cohomology of the ambient groupoid.

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References

  1. Anantharaman, C., Renault, J.: Amenable groupoids. In: Groupoids in Analysis, Geometry, and Physics (Boulder, CO, 1999), Volume 282 of Contemporary Mathematics, pp. 35–46. American Mathematical Society, Providence, RI (2001)

  2. Aschenbrenner, M., Friedl, S., Wilton, H.: 3-manifold Groups. arXiv:1205.0202 (2012)

  3. Belegradek, I.: Aspherical manifolds with relatively hyperbolic fundamental groups. Geom. Dedicata 129, 119–144 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bessières, L., Besson, G., Boileau, M., Maillot, S., Porti, J.: Geometrisation of 3-Manifolds, Volume 13 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich (2010)

  5. Bestvina, M., Fujiwara, K.: Bounded cohomology of subgroups of mapping class groups. Geom. Topol. 6, 69–89 (2002). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blank, M.: Relative Bounded Cohomology for Groupoids. PhD thesis, Universität Regensburg (2014). http://epub.uni-regensburg.de/31298/

  7. Brown, K.S.: Cohomology of Groups, Volume 87 of Graduate Texts in Mathematics. Springer, Berlin (1982)

    Google Scholar 

  8. Brown, R.: Topology and Groupoids, Third Edition of Elements of Modern Topology. BookSurge, LLC, Charleston (2006)

  9. Bucher, M., Burger, M., Frigerio, R., Iozzi, A., Pagliantini, C., Pozzetti, M.B.: Isometric Embeddings in Bounded Cohomology. arXiv:1305.2612 (2013)

  10. Bucher, M., Burger, M., Iozzi, A.: A dual interpretation of the Gromov–Thurston proof of Mostow rigidity and volume rigidity for representations of hyperbolic lattices. In: Trends in Harmonic Analysis, Volume 3 of Springer INdAM Series, pp. 47–76. Springer, Milan (2013)

  11. Bühler, T.: On the Algebraic Foundations of Bounded Cohomology. Mem. Am. Math. Soc. 214(1006), xxii+97 (2011)

  12. Chatterji, I., Fernós, T., Iozzi, A.: The Median Class and Superrigidity of Actions on Cat(0) Cube Complexes. arXiv:1212.1585 (2014)

  13. Davis, M.W., Januszkiewicz, T., Weinberger, S.: Relative hyperbolization and aspherical bordisms: an addendum to “Hyperbolization of polyhedra” [J. Differential Geom. 34 (1991), no. 2, 347–388; MR1131435 (92h:57036)] by Davis and Januszkiewicz. J. Differ. Geom. 58(3), 535–541 (2001)

  14. Franceschini, F.: A Characterization of Relatively Hyperbolic Groups Via Bounded Cohomology. arXiv:1505.04465 (2015)

  15. Frigerio, R., Pagliantini, C.: Relative measure homology and continuous bounded cohomology of topological pairs. Pac. J. Math. 257(1), 91–130 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gromov, M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. (56):5–99 (1983); 1982

  17. Hilton, P.J., Stammbach, U.: A Course in Homological Algebra, Volume 4 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1997)

  18. Ivanov, N.V.: Foundations of the theory of bounded cohomology. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 143:69–109, 177–178 (1985). Studies in topology, V

  19. Johnson, B.E.: Cohomology in Banach Algebras. American Mathematical Society, Providence, RI (1972). Memoirs of the American Mathematical Society, No. 127

  20. Kim, S., Kuessner, T.: Simplicial Volume of Compact Manifolds with Amenable Boundary. arXiv:1205.1375 (2012)

  21. Löh, C.: \(\ell ^1\)-Homology and Simplicial Volume. PhD Thesis, Universität Münster (2007). http://nbn-resolving.de/urn:nbn:de:hbz:6-37549578216

  22. Mineyev, I.: Bounded cohomology characterizes hyperbolic groups. Q. J. Math. 53(1), 59–73 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mineyev, I., Monod, N., Shalom, Y.: Ideal bicombings for hyperbolic groups and applications. Topology 43(6), 1319–1344 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Monod, N.: Continuous Bounded Cohomology of Locally Compact Groups. Lecture Notes in Mathematics, vol. 1758. Springer, Berlin (2001)

  25. Monod, N., Shalom, Y.: Cocycle superrigidity and bounded cohomology for negatively curved spaces. J. Differ. Geom. 67(3), 395–455 (2004)

    MathSciNet  MATH  Google Scholar 

  26. Munkholm, H.J.: Simplices of maximal volume in hyperbolic space, Gromov’s norm, and Gromov’s proof of Mostow’s rigidity theorem (following Thurston). In: Topology Symposium, Siegen 1979 (Proceeding of ths Symposium, Univ. Siegen, Siegen, 1979), Volume 788 of Lecture Notes in Mathematics, pp. 109–124. Springer, Berlin (1980)

  27. Noskov, G.A.: Bounded cohomology of discrete groups with coefficients (in Russian). Algebra i Analiz 2(5), 146–164 (1990)

    MathSciNet  MATH  Google Scholar 

  28. Pagliantini, C.: Relative (continuous) Bounded Cohomology and Simplicial Volume of Hyperbolic Manifolds with Geodesic Boundary. PhD thesis, Universitá di Pisa (2012). http://etd.adm.unipi.it/t/etd-07112012-101103/

  29. Park, H.: Relative bounded cohomology. Topol. Appl. 131(3), 203–234 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Paterson, A.: Amenability, Volume 29 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1988)

    Google Scholar 

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Acknowledgments

At the time of the writing of this article, the author has been supported by the SFB 1085 Higher Invariants (funded by the DFG) at the University of Regensburg. The article is based on parts of the authors Ph.D. thesis [6] at the University of Regensburg, partially supported by the GRK 1692 Curvature, Cycles and Cohomology (also funded by the DFG). The author would like to thank Francesca Diana and Cristina Pagliantini for many helpful discussions and Clara Löh for guidance and support during the project.

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  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Matthias Blank

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  1. Matthias Blank
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Blank, M. Relative bounded cohomology for groupoids. Geom Dedicata 184, 27–66 (2016). https://doi.org/10.1007/s10711-016-0156-2

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