2. Accessibility Questions
We recall the definition of strong accessibility of a group $G$ over a family of subgroups $\mathcal C$: decompose $G$ as a graph of groups with edge groups in $\mathcal C$. Then decompose the vertex groups of this graph as graphs of groups with edge groups in $\mathcal C$ and so on. If there is such a series of decompositions that terminates (i.e. at some stage the vertex groups do not admit any decomposition over $\mathcal C$) then the group is called strongly accessible over $\mathcal C$. We note that one asks only for the existence of such a series of decompositions and not that every such series terminates. This is analogous to hierarchies for 3manifolds.
Problem 2.1.
[Swarup] Let ${\mathcal C}$ be the class of finite and virtually$\mathbb Z$ groups. Are hyperbolic groups strongly accessible over ${\mathcal C}$?
Remark. Grushko’s theorem implies that finitely generated groups are strongly accessible over the trivial group. Dunwoody’s theorem [MR807066] implies that finitely presented groups are strongly accessible over ${\mathcal C}=\{\text {finite groups}\}$. Delzant and Potyagailo [MR1838998] showed that hyperbolic groups with no 2 torsion are strongly accessible over the class ${\mathcal C}$ of finite and 2ended groups. So the question remaining here is whether the no2 torsion assumption can be removed.


Problem 2.2.
[Swarup] Let ${\mathcal C}$ be the class of finite and virtually$\mathbb Z$ groups. Are finitely presented groups strongly accessible over ${\mathcal C}$?
Remark. One can ask this more generally for splittings over small groups.


Problem 2.3.
[Sageev] Is there a 1ended finitely presented group $G$ such that $G\simeq G*_{\mathbb Z}$? 
One may ask more generally
Problem 2.4.
Let ${\mathcal C}$ be the class of finitely generated groups. Are finitely presented groups strongly accessible over ${\mathcal C}$? 
Problem 2.5.
Let $G=A*_{C_n}B_n,\, n\in \mathbb N$ where $G$ is a finitely generated, $C_n$ are slender and $C_n < C_{n+1}$ for all $n$. Is it true that $G=A*_C B$ where $C < C_n$ for all $n$? Similarly for HNN extensions.
Cite this as: AimPL: JSJ Decompositions, available at http://aimpl.org/jsjdecomp.