JSJ decompositions: definitions, existence, uniqueness. I: The JSJ deformation space

TL;DR

This paper introduces a universal, canonical definition of JSJ decompositions for finitely presented groups as deformation spaces of trees, proving their existence without restrictions on edge groups and illustrating with examples.

Contribution

It provides a simple, universal definition of JSJ decompositions as deformation spaces and proves their existence for all finitely presented groups without edge group assumptions.

Findings

JSJ decompositions exist for all finitely presented groups

The decomposition is characterized as a universal maximality property

Multiple examples demonstrate the concept

Abstract

This paper and its companion arXiv:1002.4564 have been replaced by arXiv:1602.05139. We give a general simple definition of JSJ decompositions by means of a universal maximality property. The JSJ decomposition should not be viewed as a tree (which is not uniquely defined) but as a canonical deformation space of trees. We prove that JSJ decompositions of finitely presented groups always exist, without any assumption on edge groups. Many examples are given.