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## 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 3-manifolds.
1. #### Problem 2.1.

[Swarup] Let ${\mathcal C}$ be the class of finite and virtually-$\mathbb Z$ groups. Are hyperbolic groups strongly accessible over ${\mathcal C}$?
1. 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 2-ended groups. So the question remaining here is whether the no-2 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 ${\mathcal C}$?
1. Remark. One can ask this more generally for splittings over small groups.
• #### Problem 2.3.

[Sageev] Is there a 1-ended 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.
1. Remark. This was shown by Rips-Sela [MR1469317] for $C_n$ isomorphic to $\mathbb Z$. Weidmann pointed out that one may pose the question even more generally omitting ‘finitely generated’ for $G$. It is true for finitely presented groups [MR2221253].

Cite this as: AimPL: JSJ Decompositions, available at http://aimpl.org/jsjdecomp.