The homology of the Higman-Thompson groups

Abstract

We prove that Thompson's group $V$ is acyclic, answering a 1992 question of Brown in the positive. More generally, we identify the homology of the Higman-Thompson groups $V_{n,r}$ with the homology of the zeroth component of the infinite loop space of the mod $n-1$ Moore spectrum. As $V=V_{2,1}$, 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 $n$.

arXiv comment49 pages

References in corpus (8)

• Representation theory and homological stability
• Higher Dimensional Thompson Groups
• Stabilization for mapping class groups of 3-manifolds
• The Algebra of Strand Splitting. I. A Braided Version of Thompson's Group V
• Semi-simplicial spaces
• Homological stability for mapping class groups of surfaces
• The Algebra of Strand Splitting. II. A Presentation for the Braid Group on One Strand
• Representation Stability

Cited by in corpus (14)

• Homological stability for automorphism groups
• Bounded cohomology and binate groups
• Homological stability of topological moduli spaces
• The homology of the Temperley-Lieb algebras
• Homological stability for the ribbon Higman--Thompson groups
• Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces
• Asymptotic mapping class groups of Cantor manifolds and their finiteness properties
• Generalizations of Loday's assembly maps for Lawvere's algebraic theories
• On the homology of big mapping class groups
• On homological properties of the Schlichting completion
• Automorphisms of shift spaces and the Higman--Thompson groups: the one-sided case
• Stable homology of Lie algebras of derivations and homotopy invariants of wheeled operads
• Towards computing the rational homology and assembly maps of generalised Thompson groups
• Boolean algebras, Morita invariance, and the algebraic K-theory of Lawvere theories