Title: The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups (accepted for publication in K-theory, later shifted and appeared in Journal of K-theory)

Abstract: This article has two purposes. In [1] we showed that the FIC (Fibered Isomorphism Conjecture for pseudoisotopy functor) for a particular class of  3-manifolds (we denoted this class by \cal C) is the key to prove the FIC for 3-manifold groups in general. And we proved the FIC for the fundamental groups of members of a subclass of \cal C. This result was obtained by showing that the double of any member of this subclass is either Seifert fibered or supports a nonpositively curved metric. In this article we prove that for any M in \cal C there is a closed 3-manifold P such that either P is Seifert fibered or is a nonpositively curved 3-manifold and \pi_1(M) is a subgroup of \pi_1(P). As a consequence this proves that the FIC is true for any B-group (see definition 3.2 in [1]). Therefore, the FIC is true for any Haken 3-manifold group and hence for any 3-manifold group (using the reduction theorem of [1]) provided we assume the Geometrization conjecture. The above result also proves the FIC for a class of 4-manifold groups (see [12]).

            The second aspect of this article is to relax a condition in the definition of strongly poly-surface group ([9]) and define a new class of groups (we call them weak strongly poly-surface groups). Then using the above result we prove the FIC for any virtually weak strongly poly-surface group. We also give a corrected proof of  the main lemma of [9].