Cyclic splittings of finitely presented groups and the canonical JSJ deccomposition
type: Item
instance of: scholarly article
title: Cyclic splittings of finitely presented groups and the canonical JSJ deccomposition
zbMATH Open document ID: 0910.57002
DOI: 10.2307/2951832
published in: Annals of Mathematics. Second Series
publication date: 1999-01-13T00:00:00Z
review text: The general program underlying the paper is to understand groups from the point of view of low dimensional topology. In a previous paper, the second author obtained a canonical splitting of Gromov hyperbolic groups which contains, in an appropriate sense, all cyclic splittings of the group, i.e. splittings of the group into free products with amalgamation or HNN-extensions over infinite cyclic subgroups. In the present paper, this JSJ-decomposition is extended to arbitrary (freely indecomposable) finitely presented groups \(G\); here the name comes from the Jaco-Shalen-Johannson decomposition of a Haken 3-manifold into a characteristic (or Seifert fibered) part which contains all essential annuli and tori, in particular contains all splittings of the fundamental group over cyclic and rank two abelian groups, and a remaining simple or hyperbolic part). The JSJ-splitting of a Gromov hyperbolic group is unique up to a sequence of elementary modifications (slidings, conjugation and conjugation of boundary monomorphisms); this remains open for arbitrary finitely presented groups. A vocabulary is introduced which translates topological notions into algebraic ones. For example, for a separating (resp. nonseparating) simple closed curve the corresponding algebraic notion is a free product with amalgamation (resp. HNN-extension) over an infinite cyclic group. Then, like simple closed curves on a surface or a 2-orbifold, these algebraic s.c.c. may intersect or be disjoint. Given such an elementary \(\mathbb{Z}\)-splitting, an element is called elliptic if it is conjugate into a vertex group, otherwise hyperbolic. Giving two elementary \(\mathbb{Z}\)-splittings of the same group \(G\), the generator of the cyclic amalgam in one splitting may be elliptic or hyperbolic in the other splitting, so one has the three cases elliptic-elliptic, hyperbolic-hyperbolic (in which case the elementary splittings or algebraic s.c.c. are called intersecting) and elliptic-hyperbolic. One of the…
Mathematics Subject Classification ID: 57M07
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F65
zbMATH DE Number: 1192351
zbMATH Keywords: splittings over \(\mathbb{Z}\) of finitely presented groups
Wikidata QID: Q56388572
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2951832
OpenAlex ID: W4230322061
author: Eliyahu Rips
author: Zlil Sela
description: scientific article; zbMATH DE number 1192351