Regular neighbourhoods and canonical decompositions for groups.

By Peter Scott, Gadde A. Swarup. 2001

72 citations recorded by the source corpus.

We find canonical decompositions for finitely presented groups which essentially specialise to the classical JSJ-decomposition when restricted to the fundamental groups of Haken 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 set is an analogue of an immersion. This article is an announcement of work whose details will appear later. In this work, we study analogues for groups of the classical JSJ-decomposition (see Jaco and Shalen [9], Johannson [10] and Waldhausen [19]) for orientable Haken 3-manifolds. The orientability restriction is not essential but it will simplify our discussions. Previous work in this field has been carried out by Kropholler [11], Sela [18], Rips and Sela [13], Bowditch [2], [3], Dunwoody and Sageev [5], and Fujiwara and Papasoglu [8], but none of these results yields the classical JSJdecomposition when restricted to the fundamental group of an orientable Haken manifold. In our work, we give a new approach to this subject, and we give decompositions for finitely presented groups which essentially specialise to the classical JSJ-decomposition when restricted to the fundamental groups of Haken manifolds. An important feature of our approach is that the decompositions we obtain are unique and are invariant under automorphisms of the group. In previous work such strong uniqueness results were only found for decompositions of word hyperbolic groups. Most of the results of the previous authors for virtually polycyclic groups can be deduced from our work. But our arguments use some of the results of these authors, particularly those of Bowditch. In addition, we use the important work of Dunwoody and Roller in [4]. Our work also yields some extensions of the results on the Algebraic Annulus and Torus Theorems in [16], [2], and [6]. It should be remarked that even though we obtain canonical decompositions for all finitely presented groups, these decompositions are often trivial. This is analogous to the fact that any finitely generated group possesses a free product decomposition, but this decomposition is trivial whenever the given group is freely indecomposable. Our ideas are based on an algebraic generalisation of the Enclosing Property of the classical JSJ-decomposition. This property can be described briefly as follows. Received by the editors May 1, 2002, and, in revised form, July 23, 2002. 2000 Mathematics Subject Classification. Primary 20E34; Secondary 57N10, 57M07.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/32527318fdb808a63b6b7a27fe179351d4ae9aaf

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Citations in the corpus

Boundaries and JSJ Decompositions of CAT(0)-Groups (2007, methodology): “We note that the JSJ decomposition in the previous Theorem is closely related to the JSJ decomposition constructed by Scott-Swarup ([25]).”
Deformation spaces of trees (2006, background): “y are G-trees (equivalently, graphs of groups decompositions of G) with certain properties. Though canonical, they are only unique up to certain moves (see the above references, as well as [17], [4], [31]). We shall explain in [22] (see [21]) that, in general, the canonical object is not a JSJ-tree ”
Trees of cylinders and canonical splittings (2008, background): “Scott and Swarup have constructed in [26] a canonical splitting over virtually polycyclic groups.”