JournalsggdVol. 13, No. 2pp. 633–676

Acylindrical actions on trees and the Farrell–Jones conjecture

  • Svenja Knopf

    Westfälische Wilhelms-Universität Münster, Germany
Acylindrical actions on trees and the Farrell–Jones conjecture cover

A subscription is required to access this article.

Abstract

We show that for groups acting acylindrically on simplicial trees the KK- and LL-theoretic Farrell–Jones conjecture relative to the family of subgroups consisting of virtually cyclic subgroups and all subconjugates of vertex stabilisers holds. As an application, for amalgamated free products acting acylindrically on their Bass-Serre trees we obtain an identification of the associatedWaldhausen Nil-groups with a direct sum of Nil-groups associated to certain virtually cyclic groups. This identification generalises a result by Lafont and Ortiz. For a regular ring and a strictly acylindrical action these Nil-groups vanish. In particular, all our results apply to amalgamated free products over malnormal subgroups.

Cite this article

Svenja Knopf, Acylindrical actions on trees and the Farrell–Jones conjecture. Groups Geom. Dyn. 13 (2019), no. 2, pp. 633–676

DOI 10.4171/GGD/499