Splittings of Groups and Intersection Numbers

We prove algebraic analogues of the facts that a curve on a surface with self-intersection number zero is homotopic to a cover of a simple closed curve, and that two simple closed curves on a surface with intersection number zero can be isotoped to be disjoint. In [15] and [16], analogues of the classical JSJ-decompositions of Haken manifolds were described using the notion of isotopic disjointness of surfaces. The original aim of this paper was to formulate similar notions for groups. The notion of an embedded two-sided surface in a 3-manifold corresponds to a splitting of a group G over a subgroup H. The splitting defines a nontrivial H-almost invariant set X in G (see the next section for definitions). This corresponds to taking, in the universal cover of the 3-manifold, one side of a copy of the universal cover of the surface. If we also have a K-almost invariant set Y corresponding to a splitting of G over a subgroup K, the pattern of crossing (intersections) of the translates of X and Y in G leads ∗Partially supported by NSF grant DMS 034681