The homology of the Higman–Thompson groups

  • Markus SzymikEmail author
  • Nathalie Wahl


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 



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.


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Authors and Affiliations

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

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