The Symmetry of Intersection Numbers in Group Theory

For suitable subgroups of a finitely generated group, we define the intersection number of one subgroup with another subgroup and show that this number is symmetric. We also give an interpretation of this number. If one considers two simple closed curves L and S on a closed orientable surface F , one can define their intersection number to be the least number of intersection points obtainable by isotoping L and S transverse to each other. (Note that the count is to be made without any signs attached to the intersection points.) By definition, this number is symmetric, i.e. the roles of L and S are interchangeable. This can be regarded as a definition of the intersection number of the two infinite cyclic subgroups Λ and Σ of the fundamental group of F which are carried by L and S. In this paper, we show that an analogous definition of intersection number of subgroups of a group can be given in much greater generality and proved to be symmetric. We also give an interpretation of these intersection numbers. In [6], Rips and Sela considered a torsion free finitely presented groupG and infinite cyclic subgroups Λ and Σ such that G splits over each. They effectively considered the intersection number i(Λ,Σ) of Λ with Σ, and they proved that i(Λ,Σ) = 0 if and only if i(Σ,Λ) = 0. Using this, they proved that G has what they call a JSJ decomposition. If i(Λ,Σ) was not zero, it follows from their work that G can be expressed as the fundamental group of a graph of groups with some vertex group being a surface group H which contains Λ and Σ. Now it is clear (and we discuss it further in section 2 of this paper) that the intersection number of Λ with Σ is the same whether it is measured in G or in H . Also the intersection numbers of Λ and Σ in H are symmetric because of their topological interpretation. So it follows at the end of all their work that the intersection numbers of Λ and Σ in G are also symmetric. In 1994, Rips asked if there was a simpler proof of this symmetry which does not depend on their proof of the JSJ splitting. The answer is positive, and the ideas needed for the proof are all essentially contained in earlier papers of the author. This paper is a belated response to Rips’ question. The main idea is to reduce the natural, but not clearly symmetric, definition of intersection number to counting the intersections of suitably chosen sets. The most general possible algebraic situation in which to define intersection numbers seems to be that of a finitely generated group G and two subgroups Λ and Σ, not necessarily cyclic or even finitely generated, such