The Farrell–Jones conjecture for fundamental groups of graphs of abelian groups

Abstract

We show that the Farrell-Jones Conjecture holds for fundamental groups of graphs of groups with abelian vertex groups. As a special case, this shows that the conjecture holds for generalized Baumslag-Solitar groups.the Deutsche Forschungsgemeinschaft (DFG) through the Berlin Mathematical School (BMS). The

The Farrell–Jones conjecture for virtually solvable groups:

“…This result has already been used in several papers, for example, [15,16,21]. The Farrell-Jones conjecture for virtually solvable groups has been studied by several mathematicians.…”

“…Let G be a virtually solvable group. Then G satisfies the K-and L-theoretic Farrell-Jones conjecture (with coefficients in additive categories) with respect to the family of virtually cyclic subgroups.This result has already been used in several papers, for example, [15,16,21]. The Farrell-Jones conjecture for virtually solvable groups has been studied by several mathematicians.…”

Acylindrical actions on trees and the Farrell–Jones conjecture

“…For example, if two groups A and B satisfy the conjecture, then so do A × B and A * B. If A and B are abelian, then an amalgamated free product G = A * C B satisfies the Farrell-Jones Conjecture relative to VCYC by [GMR15], but for a general amalgamated free product this inheritance property is not known. In this context the last corollary can also be interpreted as follows: Any amalgamated free product G = A * C B with C being almost malnormal in either A or B acts acylindrically on its Bass-Serre tree.…”

Topological rigidity for FJ by the infinite cyclic group

“…In recent years, there has been significant progress on the proof of the Borel conjecture for a large class of groups, due to the solutions of the Farrell-Jones conjecture (FJC for short) for these groups. See the works by many authors [2], [5], [6], [7], [31], [30], [22], [18], [16], [15], [14], [28]. Roughly speaking, the conjecture says that the algebraic K-and L-groups K n (Z[G]), L n (Z[G]), n ∈ Z of the integral group ring Z[G] of a group G is determined by those of its virtually cyclic subgroups and the group homology of G. The conjecture was first formulated in [13] by Farrell and Jones with coefficients in Z.…”