# Dunwoody’s Accessibility Theorem – Day 2

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

1. H acts elliptically on T means the splitting of G does not split H (or H is contained in a vertex stabilizer).
2. H is universally elliptic if it acts elliptically on all $$\mathcal{A}$$ – trees. Example: Groups with property FA (Finite Groups).
3. An $$\mathcal {A}$$ – tree is universally elliptic, if its edges stabilizers are elliptic in every $$\mathcal {A}$$ – tree
4. 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’.
5. A JSJ decomposition (or JSJ tree) of G over  is an  – tree such that:
1. T is universally elliptic (its edge stabilizers are elliptic in every  $$\mathcal {A}$$ – tree or fixes a point in every  $$\mathcal {A}$$ – tree \)
2. T dominates any other universally elliptic tree T’ (the vertex stabilizers are as small as possible; they are elliptic in every universally elliptic tree).
3. 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.