JSJ decompositions of groups

Vincent Guirardel, Gilbert Levitt

Published 2016-02-16Version 1

This is an account of the theory of JSJ decompositions of finitely generated groups, as developed in the last twenty years or so. We give a simple general definition of JSJ decompositions (or rather of their Bass-Serre trees), as maximal universally elliptic trees. In general, there is no preferred JSJ decomposition, and the right object to consider is the whole set of JSJ decompositions, which forms a contractible space: the JSJ deformation space. We prove that JSJ decompositions exist for any finitely presented group $G$, without any assumption on edge groups. When edge groups are slender, we describe flexible vertices of JSJ decompositions as quadratically hanging extensions of 2-orbifold groups. Similar results hold in the presence of acylindricity, in particular for splittings of torsion-free CSA groups over abelian groups, and splittings of relatively hyperbolic groups over virtually cyclic or parabolic subgroups. Using trees of cylinders, we obtain canonical JSJ trees (which are invariant under automorphisms). We introduce the compatibility JSJ decomposition, replacing the property of being universally elliptic by the stronger property of being universally compatible. This yields a canonical compatibility JSJ tree, not just a deformation space. We show that it exists when $G$ is finitely presented. We give many examples, and we work throughout with relative decompositions (certain subgroups are required to be elliptic in all trees under consideration).