Canonical splittings of groups and 3-manifolds
The standard JSJ-decomposition of an orientable Haken 3-manifold is described in terms that can be translated into group theory and a purely algebraic description of it is given. This leads to a natural algebraic proof of Johannson's Deformation Theorem [\textit{K. Johannson}, Homotopy equivalences of 3-manifolds with boundaries, Lect. Notes Math. 761 (1979; Zbl 0412.57007)]. In the first section, the authors give the background material on 3-manifolds and group theory. In the second section analogues of canonic annuli and tori are studied, namely canonical \(\mathbb{Z}\) or \(\mathbb{Z}\times \mathbb{Z}\) splittings of \(G=\pi _{1}(M)\). Since this characterization is algebraic, it follows that the JSJ-systems of any two Haken manifolds with isomorphic fundamental groups correspond. The last section deals with the result that we mentioned above.
scientific article; zbMATH DE number 1663174
scientific article