"Revisiting Johannson’s Deformation Theorem"

Abstract:  Jaco-Shalen and Johannson developed in 1979, what are now called JSJ decompositions, for Haken manifolds with incompressible boundary. This decomposes the three manifold into Canonical simple pieces and fibered pieces. Johannson also proved that any homotopy equivalence between two such Haken manifolds can be deformed to carry simple pieces to simple pieces and fibered pieces to fibered pieces. Scott and Swarup developed recently a similar decomposition for Poincare Duality pairs of dimensions n+2, n>0 (think of these as aspherical manifolds with incompressible boundary). We show now that any isomorphism in the fundamental groups  between two Poincare Duality pairs has the same property, that is, it is independent of the boundary structure. The theory involves the so called almost invariant sets over virtually polycyclic groups, which correspond to immersions over annuli and tori in the three dimensional case. This is joint work with Lawrence Reeves and Peter Scott.