Journal of the Mathematical Society of Japan

A note on bounded-cohomological dimension of discrete groups

Clara LÖH

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Bounded-cohomological dimension of groups is a relative of classical cohomological dimension, defined in terms of bounded cohomology with trivial coefficients instead of ordinary group cohomology. We will discuss constructions that lead to groups with infinite bounded-cohomological dimension, and we will provide new examples of groups with bounded-cohomological dimension equal to 0. In particular, we will prove that every group functorially embeds into an acyclic group with trivial bounded cohomology.

Article information

Source
J. Math. Soc. Japan Volume 69, Number 2 (2017), 715-734.

Dates
First available in Project Euclid: 20 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1492653644

Digital Object Identifier
doi:10.2969/jmsj/06920715

Subjects
Primary: 55N35: Other homology theories 20J06: Cohomology of groups 20E99: None of the above, but in this section

Keywords
bounded cohomology cohomological dimension mitotic groups

Citation

LÖH, Clara. A note on bounded-cohomological dimension of discrete groups. J. Math. Soc. Japan 69 (2017), no. 2, 715--734. doi:10.2969/jmsj/06920715. https://projecteuclid.org/euclid.jmsj/1492653644


Export citation

References

  • G. Baumslag, E. Dyer and A. Heller, The topology of discrete groups, J. Pure Appl. Algebra, 16 (1980), 1–47.
  • A. Bouarich, Théorèmes de Zilber-Eilemberg et de Brown en homologie $\ell^1$, Proyecciones, 23 (2004), 151–186.
  • T. Bühler, On the algebraic foundations of bounded cohomology, Mem. Amer. Math. Soc., 214 (2011), no. 1006.
  • A. Dold, Lectures on Algebraic Topology, reprint of the 1972 edition, Classics in Mathematics, Springer, 1995.
  • K. Fujiwara, The second bounded cohomology of an amalgamated free product of groups, Trans. Amer. Math. Soc., 352 (2000), 1113–1129.
  • R. I. Grigorchuk, Some results on bounded cohomology, Combinatorial and geometric group theory (Edinburgh, 1993), London Math. Soc. Lecture Note Ser., 204, Cambridge University Press, 1995, 111–163.
  • R. I. Grigorchuk, Bounded cohomology of group constructions, Mat. Zametki, 59 (1996), 546–550; translation in Math. Notes, 59 (1996), 392–394.
  • M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math., 56 (1983), 5–99.
  • T. Hartnick and A. Ott, Bounded cohomology via partial differential equations, I, preprint, arXiv:1310.4806 [math.GR], 2013.
  • H. Inoue and K. Yano, The Gromov invariant of negatively curved manifolds, Topology, 21 (1981), 83–89.
  • N. V. Ivanov, Foundations of the theory of bounded cohomology, J. Soviet Math., 37 (1987), 1090–1114.
  • C. Löh, Isomorphisms in $\ell^1$-homology, Münster J. of Math., 1 (2008), 237–266.
  • C. Löh and R. Sauer, Simplicial volume of Hilbert modular varieties, Comment. Math. Helv., 84 (2009), 457–470.
  • C. Löh, Simplicial Volume, Bull. Man. Atl., 2011, 7–18.
  • J. N. Mather, The vanishing of the homology of certain groups of homeomorphisms, Topology, 10 (1971), 297–298.
  • S. Matsumoto and S. Morita, Bounded cohomology of certain groups of homeomorphisms, Proc. Amer. Math. Soc., 94 (1985), 539–544.
  • I. Mineyev, Straightening and bounded cohomology of hyperbolic groups, Geom. Funct. Anal., 11 (2001), 807–839.
  • Y. Mitsumatsu, Bounded cohomology and $l^1$-homology of surfaces, Topology, 23 (1984), 465–471.
  • N. Monod, Continuous Bounded Cohomology of Locally Compact Groups, Lecture Notes in Math., 1758, Springer, 2001.
  • N. Monod, An invitation to bounded cohomology, International Congress of Mathematicians, II, Eur. Math. Soc., 2006., 1183–1211.
  • T. Soma, Bounded cohomology of closed surfaces, Topology, 36 (1997), 1221–1246.
  • T. Soma, Existence of non-Banach bounded cohomology, Topology, 37 (1998), 179–193.
  • W. P. Thurston, The geometry and topology of $3$-manifolds, mimeographed notes, 1979.
  • T. Yoshida, On $3$-dimensional bounded cohomology of surfaces, Homotopy theory and related topics (Kyoto, 1984), Adv. Stud. Pure Math., 9, North-Holland, 1987, 173–176.