JSJ decompositions: definitions, existence, uniqueness. II. Compatibility and acylindricity

TL;DR

This paper introduces the compatibility JSJ tree for groups, proves its existence under certain conditions, and explores its properties and applications to various classes of groups, enhancing understanding of group decompositions.

Contribution

It defines the compatibility JSJ tree, proves its existence under finitely presented and acylindricity conditions, and analyzes its properties and applications.

Findings

Compatibility JSJ tree exists for finitely presented groups.

Under acylindricity, the JSJ deformation space and compatibility JSJ tree exist.

The flexible subgroups of these trees are characterized.

Abstract

This paper and its companion arXiv:0911.3173 have been replaced by arXiv:1602.05139. We define the compatibility JSJ tree of a group G over a class of subgroups. It exists whenever G is finitely presented and leads to a canonical tree (not a deformation space) which is invariant under automorphisms. Under acylindricity hypotheses, we prove that the (usual) JSJ deformation space and the compatibility JSJ tree exist, and we describe their flexible subgroups. We apply these results to finitely generated CSA groups, \Gamma-limit groups (allowing torsion), and relatively hyperbolic groups.