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Splittings of groups and intersection numbers. (English) Zbl 0983.20024
In a previous paper [Geom. Topol. 2, 11-29 (1998; Zbl 0897.20029)], the first author defined the intersection number of two splittings (as an amalgamated free product or HNN extension) of any group \(G\) over any subgroups \(H\) and \(K\). In the special case when \(G\) is the fundamental group of a compact surface \(F\) and these splittings arise from embedded arcs or circles on \(F\), the algebraic intersection number of the splittings equals the topological intersection numbers of the corresponding 1-manifolds. The analogous statement holds when \(G\) is the fundamental group of a compact 3-manifold and these splittings arise from \(\pi_1\)-injective embedded surfaces.
“The first main result of the paper is a generalisation to the algebraic setting of the fact that two simple arcs or closed curves on a surface have intersection number zero if and only if they can be isotoped apart.” Here the algebraic analogue of isotopy is defined in terms of compatibility of splittings which roughly means that the group \(G\) can be expressed as the fundamental group of a graph of groups realizing simultaneously the given splittings. Now the first main theorem says that a collection of \(n\) splittings over finitely generated subgroups is compatible up to isotopy if and only if each pair of splittings has intersection number zero; also, uniqueness of the underlying graph and of its edge and vertex groups is obtained. Next an algebraic analogue of non-embedded arcs or circles on surfaces resp. of immersed \(\pi_1\)-injective surfaces in 3-manifolds is discussed which is defined in terms of almost invariant subsets of the quotient \(H\setminus G\), for a subgroup \(H\) of \(G\).
“Our second main result is an algebraic analogue of the fact that a singular curve on a surface or a singular surface in a 3-manifold which has self-intersection number zero can be homotoped to cover an embedding. It asserts that if \(H\setminus G\) has an almost invariant subset with self-intersection number zero, then \(G\) has a splitting over a subgroup commensurable with \(H\).”
“In a separate paper, we use the ideas about intersection numbers of splittings to study the JSJ decomposition of Haken 3-manifolds. The problem is to recognize which splittings of the fundamental group of such a manifold arise from the JSJ decomposition. It turns out that a class of splittings which we call canonical can be defined using intersection numbers, and we use this to show that the JSJ decomposition for Haken 3-manifolds depends only on the fundamental group. This leads to an algebraic proof of Johannson’s Deformation Theorem. It seems very likely that similar ideas apply to Z. Sela’s JSJ decomposition of hyperbolic groups [Geom. Funct. Anal. 7, No. 3, 561-593 (1997; Zbl 0884.20025)] and thus provide a common thread to the two types of JSJ decompositions. Thus, the use of intersection numbers seems to provide a tool in the study of diverse topics in group theory, and this paper together with the paper of Scott mentioned above provides some of the foundational material”.

MSC:
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F65 Geometric group theory
57M07 Topological methods in group theory
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