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Zbl 1036.20028
Scott, Peter; Swarup, Gadde A.
Regular neighbourhoods and canonical decompositions for groups.
[B] Astérisque 289. Paris: Société Mathématique de France. vi, 234~p. \$~58.00; EUR~40.00 (2003). ISBN 2-85629-146-5/pbk

Regular neighbourhoods (of embedded or immersed submanifolds) are one of the basic tools in manifold theory. The present paper gives a version of this notion for groups, with the main intent to produce an analogue for arbitrary finitely presented groups of the Jaco-Shalen-Johannson decomposition of Haken 3-manifolds (along tori and annuli into simple or hyperbolic 3-manifolds and Seifert fiber spaces), resp. of the corresponding decomposition of their fundamental groups into graphs of groups. The characteristic submanifold of a Haken 3-manifold $M$ consists of the Seifert fiber pieces of the decomposition and can be thought of as regular neighbourhood of all the essential (embedded or immersed) tori and annuli in $M$. So one is looking for a decomposition of a group $G$ as a graph of groups, with Abelian edge groups and a subset of characteristic vertex groups which contains conjugates of certain essential subgroups of the group (such as free groups of rank two). There are various previous approaches to such a decomposition which, however, either do not specialize to the JSJ-decomposition in the case of fundamental groups of Haken 3-manifolds, or have the strong uniqueness properties of the JSJ-decomposition only for certain classes of groups (e.g. word hyperbolic groups) (see the introduction of the present paper for a careful description of the various previous approaches to such a decomposition).\par ``We find a canonical decomposition for (almost) finitely presented groups which essentially specialize to the classical Jaco-Shalen-Johannson decomposition when restricted to the fundamental groups of Haken 3-manifolds. The decompositions that we obtain are invariant under automorphisms of the group. A crucial new ingredient is the concept of a regular neighbourhood of a family of almost invariant subsets of a group. An almost invariant subset is an analogue of an immersion."\par ``Here is an introduction to our ideas. As mentioned above, our choice of the crucial property of the classical JSJ-decomposition is the Enclosing Property for immersions. This property implies that the characteristic submanifold $V(M)$ of an orientable Haken 3-manifold $M$ contains a representative of every homotopy class of an essential annulus or torus in $M$. In this paper, we introduce a natural algebraic analogue of enclosing." ``Let $\sigma$ be a splitting of $G$. We say that $\sigma$ is enclosed by a vertex $v$ of a graph of groups structure $\Gamma$ of $G$, if there is a graph of groups structure $\Gamma_\sigma$ of $G$, with an edge $e$ which determines the splitting $\sigma$, such that collapsing the edge $e$ yields $\Gamma$, and $v$ is the image of $e$. We emphasize that the condition that $\sigma$ is enclosed by a vertex $v$ is in general stronger than the condition that the edge group of $\sigma$ is conjugate into the vertex group of $v$. This is particularly clear if $\sigma$ is a free product decomposition of $G$, as then the edge group of $\sigma$ is trivial."\par Previous results of the authors' on algebraic analogues of the fact that curves on surfaces with intersection number zero can be homotoped to be disjoint suggest that an appropriate algebraic analogue of an immersion is an almost invariant subset of $G$ ([Geom. Topol. 4, 179-218 (2000; Zbl 0983.20024)], contained as an appendix in the present paper). Thinking of the characteristic submanifold as a regular neighbourhood of all the essential annuli and tori in $M$, ``the key new idea of the paper is an algebraic version of regular neighbourhood theory. We describe an algebraic regular neighbourhood of a family of almost invariant subsets of $G$. This is a graph of groups structure for $G$, with the property that certain vertices enclose the given almost invariant sets. As splittings have almost invariant sets naturally associated, this also yields an idea of an algebraic regular neighbourhood of a family of splittings."
[Bruno Zimmermann (Trieste)]
MSC 2000:
*20E06 Free products and generalizations (group theory)
20F65 Geometric group theory
57M07 Topological methods in group theory

Keywords: decompositions of groups as graphs of groups; JSJ-decompositions; finitely presented groups; fundamental groups; free products

Citations: Zbl 0983.20024

Cited in: Zbl 1282.20047 Zbl 1234.20032

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