In this note, we give an alternate proof of the Farrell–Jones isomorphism conjecture for the affine Artin groups of type $\widetilde B_n$.

We observe an inductive structure in a large class of Artin groups of finite real, complex and affine types and exploit this information to deduce the Farrell–Jones isomorphism conjecture for these groups.

Let $k$ be a field of characteristic 0. Let $G$ be a reductive group over the ring of Laurent polynomials $R\,=\,k\left[ x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1} \right]$ . Assume that $G$ contains a maximal $R$ -torus, and that every semisimple normal subgroup of $G$ contains a two-dimensional split torus $\mathbf{G}_{m}^{2}$ . We show that the natural map of non-stable ${{K}_{1}}$ -functors, also called Whitehead groups, $K_{1}^{G}\left( R \right)\,\to \,K_{1}^{G}\left( k\left( \left( {{x}_{1}} \right) \right)\cdots \left( \left( {{x}_{n}} \right) \right) \right)$ is injective, and an isomorphism if $G$ is semisimple. As an application, we provide a way to compute the difference between the full automorphism group of a Lie torus (in the sense of Yoshii–Neher) and the subgroup generated by exponential automorphisms.

Let G be a reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank ≥ 2, i.e. contains (Gm)2. Let K1G be the non-stable K1-functor associated to G, also called the Whitehead group of G. We show that K1G(k) = K1G (k[X1 ,…, Xn]) for any n ≥ 1. If k is perfect, this implies that K1G (R) = K1G (R[X]) for any regular k-algebra R. If k is infinite perfect, one also deduces that K1G (R) → K1G (K) is injective for any local regular k-algebra R with the fraction field K.

We show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for fundamental groups of a large class of 3-manifolds. We also prove that if the FIC is true for irreducible 3-manifold groups then it is true for all 3-manifold groups. In fact, this follows from a more general result we prove, namely we show that if the FIC is true for each vertex group of a graph of groups with trivial edge groups then the FIC is true for the fundamental group of the graph of groups. This result is part of a program to prove the FIC for the fundamental group of a graph of groups where all the vertex and edge groups satisfy the FIC. A consequence of the first result gives a partial solution to a problem in the problem list of R. Kirby. We also deduce that the FIC is true for a class of virtually P D3-groups.

Another main aspect of this article is to prove the FIC for all Haken 3-manifold groups assuming that the FIC is true for B-groups. By definition a B-group contains a finite index subgroup isomorphic to the fundamental group of a compact irreducible 3-manifold with incompressible nonempty boundary so that each boundary component is of genus ≥ 2. We also prove the FIC for a large class of B-groups and moreover, using a recent result of L.E. Jones we show that the surjective part of the FIC is true for any B-group.

We first prove that the Whitehead group of a torsion-free virtually solvable linear group vanishes. Next we make a reduction of the fibered isomorphism conjecture from virtually solvable groups to a class of virtually solvable $\mathbb {Q}$-linear groups. Finally we prove an L-theory analogue for elementary amenable groups.