JSJ Decompositions/Section2 - Problems in Geometric Group Theory

2 JSJ decompositions

Problem 2.1 (Comments) (Rips) Is there a JSJ decomposition for finitely presented groups over small groups?

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 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. Dunwoody Du 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 MR2319455, GL.

Problem 2.2 (Comments) (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 ${\mathbb F}_2$ ?

Remark We call a splitting of $G$ over $C$ unfolded if $G$ does not split over any proper subgroup of $C$ .

A splitting of a group $G$ over a group $C_1$ is called 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 ( 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, ${\mathbb F}_2$ .

Problem 2.3 (Comments) (Levitt) Is the isomorphism problem solvable for the graphs of groups for which all edges and vertices are labelled by $\mathbb Z$ ?

Remark One of the main applications of the JSJ theory was the solution of the isomorphism problem for hyperbolic groups. In general however there is no canonical JSJ decomposition (see though MR2032389) and one does not have a good description of the set of all JSJ decompositions. The simplest case for which this question is open is that of graphs of groups where all vertex and edge groups are isomorphic to $\mathbb Z$ . In this case an algorithmic description is equivalent to the isomorphism problem stated above. Levitt remarks that the isomorphism problem for graphs of groups where all edges and vertices are labelled by $\mathbb Z ^4$ is unsolvable.

Problem 2.4 (Comments) (Swarup) Let $G$ be a $CAT(0)$ group with $Out(G)$ infinite. Is it true that $G$ admits a Dehn twist of infinite order?

Remark A Dehn twist of $G$ is an automorphism $\phi$ of the following form: either $G$ splits as $A*_CB$ , $t\in C$ is central, $\phi (a)=a$ for $a\in A$ and $\phi (b)=tbt^{-1}$ for $b\in B$ ; or $G$ is an HNN extension $A*_C$ , with stable letter $s$ , $t\in C$ is central, $\phi (a)=a$ for $a\in A$ and $\phi (s)=ts$ .

It was shown by Levitt MR2174093 that a one ended hyperbolic group with $Out(G)$ infinite admits a Dehn twist of infinite order coming from a splitting over a virtually cyclic group.

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