The Farrell-Jones isomorphism conjecture for 3-manifold groups

Abstract

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.

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

References in corpus (5)

• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
• The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups
• K-theory of virtually poly-surface groups
• The fibered isomorphism conjecture for complex manifolds
• Controlled K-theory I: Basic theory

Cited by in corpus (14)

• The Farrell-Jones Conjecture for virtually solvable groups
• Algebraic K-theory of group rings and the cyclotomic trace map
• The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups
• On the isomorphism conjecture for groups acting on trees
• On the Farrell-Jones Conjecture for Waldhausen's $A$-theory
• The isomorphism conjecture in L-theory: graphs of groups
• Algebraic K-theory of groups wreath product with finite groups
• Vanishing structure set of 3-manifolds
• K- and L-theory of group rings over GL_n(Z)
• Nilpotent invariance of semi-topological K-theory of dg-algebras and the lattice conjecture
• On the Farrell-Jones and related Conjectures
• The isomorphism conjecture for groups with generalized free product structure
• The isomorphism conjecture in L-theory
• The isomorphism conjecture for Artin groups