JSJ decomposition, also known as the toral decomposition

JSJ decomposition is a very important and effective tool in the research process.

JSJ decompositions of groups

In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem. Toral decomposition is the diagonalization of the gradient matrix. The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. In this sequence of papers, we study the automorphisms of a hierarchically hyperbolic group that satisfies some weak acylindricity conditions. To study these automorphisms we construct an object that can be viewed as a higher rank JSJ decomposition. In the first paper we demonstrate our construction in the case of a right angled Artin group. For studying automorphisms of a general HHG we construct what we view as a higher rank Makanin-Razborov diagram, which is the first step in the construction of the higher rank JSJ. For example, Using JSJ-decomposition, we can express the fundamental group π1(M) of an irreducible connected orientable 3-manifold M as a graph of groups G.

JSJ decompositions of finitely generated groups are a fundamental tool in geometric group theory. In the second paper we use this diagram, and techniques from the solution to Tarski's problem on the elementary theory of a free group, to associate two groupoids with this automorphism group. The groupoids generalize the structure of the automorphism group of a hyperbolic group that follows from the existence of its JSJ decomposition. In this paper we construct the groupoids for HHGs with single ended higher rank MR diagrams. In the next paper of the sequence we generalize the construction to HHGs with general higher rank diagrams. We classify all possible JSJ decompositions of doubles of free groups of rank two, and we also compute the Makanin–Razborov diagram of a particular double of a free group and deduce that in general limit groups are not freely subgroup separable.

JSJ decomposition References

Abelian splittings and JSJ-decompositions of finitely presented Bestvina–Brady groups

YC Chang - Journal of Group Theory, 2023

Boundaries and JSJ decompositions of CAT (0)-groups

P Papasoglu, E Swenson - Geometric And Functional Analysis, 2009

JSJ-decompositions of finitely presented groups and complexes of groups

K Fujiwara, P Papasoglu - Geometric & Functional Analysis GAFA, 2006