The Farrell-Jones isomorphism conjecture for 3-manifold groups

Details The Farrell-Jones isomorphism conjecture for 3-manifold groups The Farrell-Jones isomorphism conjecture for 3-manifold groups

2004-05-12

arxiv.org/abs/math/0405211v4

J. K-Theory 1 (2008) 49-82

Mathematics

K-Theory and Homology

35 pages, 1 figure (.eps file), AMS Latex file, final version. accepted for publication in K-theory

Scientific paper

10.1017/is007011012jkt005

We show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for the 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 here, 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 FIC for the fundamental group of a graph of groups where all the vertex and edge groups satisfy 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 PD_3-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 \geq 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.

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