Advertisement

The homology of the Higman–Thompson groups

  • Markus SzymikEmail author
  • Nathalie Wahl
Article
  • 13 Downloads

Abstract

We prove that Thompson’s group \(\mathrm {V}\) is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman–Thompson groups \(\mathrm {V}_{n,r}\) with the homology of the zeroth component of the infinite loop space of the mod \(n-1\) Moore spectrum. As \(\mathrm {V}=\mathrm {V}_{2,1}\), we can deduce that this group is acyclic. Our proof involves establishing homological stability with respect to r, as well as a computation of the algebraic K-theory of the category of finitely generated free Cantor algebras of any type n.

Mathematics Subject Classification

19D23 20J05 

Notes

Acknowledgements

This research has been supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92) in Copenhagen. Parts of this paper were conceived while the authors were visiting the Hausdorff Research Institute for Mathematics (HIM) in Bonn, and the paper was revised during a visit at the Newton Institute in Cambridge (EPSRC Grants EP/K032208/1 and EP/R014604/1). We thank both institutes for their support. The authors would like to thank Dustin Clausen and Oscar Randal-Williams for pointing out gaps in early versions of this paper, and the referee for a report that helped us improving the paper. The first author would also like to thank Ricardo Andrade, Ken Brown, Bjørn Dundas, Magdalena Musat, Martin Palmer, and Vlad Sergiescu for conversations related to the subject of this paper.

References

  1. 1.
    Arlettaz, D.: The first \(k\)-invariant of a double loop space is trivial. Arch. Math. 54, 84–92 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barratt, M., Priddy, S.: On the homology of non-connected monoids and their associated groups. Comment. Math. Helv. 47, 1–14 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brown, K.S.: Finiteness properties of groups. J. Pure Appl. Algebra 44, 45–75 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brown, K.S.: The Geometry of Finitely Presented Infinite Simple Groups. Algorithms and Classification in Combinatorial Group Theory (Berkeley, CA, 1989). Mathematical Sciences Research Institute Publications, vol. 23, pp. 121–136. Springer, New York (1992)Google Scholar
  5. 5.
    Brown, K.S., Geoghegan, R.: An infinite-dimensional torsion-free \(\text{ FP }_\infty \) group. Invent. Math. 77, 367–381 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ebert, J., Randal-Williams, O.: Semi-simplicial spaces. Preprint (2017). arXiv:1705.03774
  7. 7.
    Farley, D.S.: Homological and finiteness properties of picture groups. Trans. Am. Math. Soc. 357, 3567–3584 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ghys, É., Sergiescu, V.: Sur un groupe remarquable de difféomorphismes du cercle. Comment. Math. Helv. 62, 185–239 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grayson, D.: Higher Algebraic K-theory. II (After Daniel Quillen). Algebraic K-theory (Proceedings of Conference, Northwestern University, Evanston, Ill, 1976). Lecture Notes in Mathematics, vol. 551, pp. 217–240. Springer, Berlin (1976)Google Scholar
  10. 10.
    Hatcher, A., Wahl, N.: Stabilization for mapping class groups of 3-manifolds. Duke Math. J. 155, 205–269 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Higman, G.: Finitely Presented Infinite Simple Groups. Notes on Pure Mathematics 8. Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra (1974)Google Scholar
  12. 12.
    Kapoudjian, C.: Virasoro-type extensions for the Higman–Thompson and Neretin groups. Q. J. Math. 53, 295–317 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    McDuff, D., Segal, G.: Homology fibrations and the “group-completion” theorem. Invent. Math. 31, 279–284 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mukai, J.: Stable homotopy of some elementary complexes. Mem. Fac. Sci. Kyushu Univ. Ser. A 20, 266–282 (1966)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Quillen, D.: Higher Algebraic K-theory. I. Algebraic K-theory, I: Higher K-Theories (Proceedings of the Conference, Battelle Memorial Institute, Seattle, Washington, 1972). Lecture Notes in Mathematics, vol. 341, pp. 85–147. Springer, Berlin (1973)Google Scholar
  16. 16.
    Randal-Williams, O.: ‘Group-completion’, local coefficient systems and perfection. Q. J. Math. 64, 795–803 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Randal-Williams, O., Wahl, N.: Homological stability for automorphism groups. Adv. Math. 318, 534–626 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Thomason, R.W.: First quadrant spectral sequences in algebraic K-theory via homotopy colimits. Commun. Algebra 10, 1589–1668 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Thumann, W.: Operad groups and their finiteness properties. Adv. Math. 307, 417–487 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zeeman, E.C.: Relative simplicial approximation. Proc. Camb. Philos. Soc. 60, 39–43 (1964)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNTNU Norwegian University of Science and TechnologyTrondheimNorway
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations