Problem 1.1 [Rips] Is there a JSJ decomposition for finitely presented groups over small groups?

Remark. Dunwoody has announced recently a much more general result for groups acting on $\mathbb R$-trees that seems to imply the existence of JSJ decompositions over small groups. Guirardel and Levitt have also proposed a generalized JSJ theory in the spirit of outer space \cite{MR2319455}.

Remark. The first case to look at is the case of decompositions over solvable Baumslag-Solitar groups. If such a decomposition exists then the edge groups of the decomposition won't necessarily be small (see \cite{MR2221253}). On the other hand there is a natural conjecture for the enclosing groups: one expects them to be fundamental groups of complexes of groups where the underlying complex is a square complex homeomorphic to a surface and all edges and faces are stabilized by a fixed group $F$ (the fiber). However the homomorphisms from edge groups to face groups are not necessarily isomorphisms.

We call a splitting of $G$ over $C$ \emph{unfolded} if $G$ does not split over any proper subgroup of $C$.

Problem 1.2 [Panos Papasoglu] Let $G$ be a finitely presented group that does not split over a virtually abelian group. Can $G$ have infinitely many unfolded splittings over distinct subgroups isomorphic to ${\mathbb F}_2$?

This is just a test status statement

Remark. A splitting of a group $G$ over a group $C_1$ is called \emph{elliptic} with respect to a splitting of $G$ over $C_2$ if $C_1$ fixes a point of the Bass-Serre tree of the splitting over $C_2$. Otherwise it is called hyperbolic.

A pair of splittings of $G$ over $C_1,C_2$ can be elliptic-elliptic, elliptic-hyperbolic, hyperbolic-elliptic, hyperbolic-hyperbolic.

A crucial observation for the JSJ theory of splittings of 1-ended finitely presented groups over $\mathbb Z$ is that any two splittings over infinite cyclic groups $C_1,C_2$ are either elliptic-elliptic or hyperbolic-hyperbolic.

The Bestvina-Feighn accessibility theorem (\cite{MR1091614}) gives a bound on the number of elliptic-elliptic splittings over $\mathbb Z$ (and more generally over small groups). So the main issue for the JSJ theory is understanding hyperbolic-hyperbolic splittings. In the case of $\mathbb Z$-splittings it is easy to produce infinitely many distinct hyperbolic-hyperbolic splittings, just consider the splittings corresponding to simple closed curves on a surface. The JSJ theory says that in fact this is the only way in which such splittings arise. So the problem above asks whether there are families of hyperbolic-hyperbolic splittings over the free group of rank 2, ${\mathbb F}_2$.