# Acylindrical Actions on Trees and the Farrell-Jones Conjecture

## TL;DR

This paper proves the Farrell-Jones Conjecture for groups acting acylindrically on trees, providing new insights into algebraic K- and L-theory, and generalizing previous results on Nil-groups for amalgamated free products.

## Contribution

It establishes the Farrell-Jones Conjecture for acylindrical actions on trees and identifies associated Nil-groups with those of virtually cyclic groups, extending prior work.

## Key findings

- Farrell-Jones Conjecture holds for acylindrical actions on trees.
- Nil-groups associated with amalgamated free products can be explicitly identified.
- Nil-groups vanish for regular rings and strictly acylindrical actions.

## Abstract

We show that for groups acting acylindrically on simplicial trees the $K$- and $L$-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 associated Waldhausen Nil-groups with a direct sum of Nil-groups associated to certain virtually cyclic groups. This identification generalizes 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.