Suppose G is a finitely presented group. Let us fix \( \mathcal{A} \) – a favorite class of subgroups of G (closed under taking subgroups and conjugation).

If G acts on an \(\mathcal{A} \) – tree T, we have a graph of groups decomposition for G.

If H < G then H acts on T and we have graph of groups decomposition for H.

## Definitions

- H
**acts elliptically**on T means the splitting of G does not split H (or H is contained in a vertex stabilizer). - H
**is universally elliptic**if it acts elliptically on all \( \mathcal{A} \) – trees. Example: Groups with property FA (Finite Groups). - An \( \mathcal {A} \) – tree is universally elliptic, if its edges stabilizers are elliptic in every \(\mathcal {A} \) – tree
**T dominates T’**means there exists a G-equivariant map from T to T’. This further implies that if some subgroup H is not split by T then H is not split by T’.**A JSJ decomposition (or JSJ tree)**of G over is an – tree such that:- T is universally elliptic (its edge stabilizers are elliptic in every \(\mathcal {A} \) – tree or fixes a point in every \(\mathcal {A} \) – tree \)
- T dominates any other universally elliptic tree T’ (the vertex stabilizers are as small as possible; they are elliptic in every universally elliptic tree).
**Example:**If \(\mathcal {A} \) only contains the trivial group, JSJ trees are same as Grushko trees.

## Motivation from JSJ Decomposition of compact 3 manifold

William Jaco, Peter Shalen, and Klaus Johannson (1979) proved the following:

Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered.

### Comparing JSJ decomposition of groups and manifold

Graph of groups decomposition of G over \( \mathcal{A} \).

Graph of groups for H < G

H acts elliptically on T

H is universally elliptic

JSJ decomposition of a 3 manifold by cutting along embedded family of tori

H ‘cut’ by the family of tori.

Tori can be pushed off of H

H is not cut by any embedded torus

## Goal

We wish to show that for every sequence of graph of groups decomposition of a finitely presented group G, over a favored class of subgroups \( \mathcal {A} \), there exists a JSJ decomposition that dominates all decompositions eventually.