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
dvipng error! exitcode was 2 (signal 0), transscript follows:(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
dvipng error! exitcode was 2 (signal 0), transscript follows:-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
dvipng error! exitcode was 2 (signal 0), transscript follows:be a finitely presented group that does not split over a virtually abelian group. Can
dvipng error! exitcode was 2 (signal 0), transscript follows:have infinitely many unfolded splittings over distinct subgroups isomorphic to
dvipng error! exitcode was 2 (signal 0), transscript follows:?
Remark We call a splitting of
dvipng error! exitcode was 2 (signal 0), transscript follows:over
dvipng error! exitcode was 2 (signal 0), transscript follows:unfolded if
dvipng error! exitcode was 2 (signal 0), transscript follows:does not split over any proper subgroup of
dvipng error! exitcode was 2 (signal 0), transscript follows:.
A splitting of a group
dvipng error! exitcode was 2 (signal 0), transscript follows:over a group
dvipng error! exitcode was 2 (signal 0), transscript follows:is called elliptic with respect to a splitting of
dvipng error! exitcode was 2 (signal 0), transscript follows:over
dvipng error! exitcode was 2 (signal 0), transscript follows:if
dvipng error! exitcode was 2 (signal 0), transscript follows:fixes a point of the Bass-Serre tree of the splitting over
dvipng error! exitcode was 2 (signal 0), transscript follows:. Otherwise it is called hyperbolic.
A pair of splittings of
dvipng error! exitcode was 2 (signal 0), transscript follows:over
dvipng error! exitcode was 2 (signal 0), transscript follows: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
dvipng error! exitcode was 2 (signal 0), transscript follows:is that any two splittings over infinite cyclic groups
dvipng error! exitcode was 2 (signal 0), transscript follows:are either elliptic-elliptic or hyperbolic-hyperbolic.
The Bestvina-Feighn accessibility theorem ( MR1091614) gives a bound on the number of elliptic-elliptic splittings over
dvipng error! exitcode was 2 (signal 0), transscript follows:(and more generally over small groups). So the main issue for the JSJ theory is understanding hyperbolic-hyperbolic splittings. In the case of
dvipng error! exitcode was 2 (signal 0), transscript follows:-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,
dvipng error! exitcode was 2 (signal 0), transscript follows:.
Problem 2.3 (Comments) (Levitt) Is the isomorphism problem solvable for the graphs of groups for which all edges and vertices are labelled by
dvipng error! exitcode was 2 (signal 0), transscript follows:?
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
dvipng error! exitcode was 2 (signal 0), transscript follows:. 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
dvipng error! exitcode was 2 (signal 0), transscript follows:is unsolvable.
Problem 2.4 (Comments) (Swarup) Let
dvipng error! exitcode was 2 (signal 0), transscript follows:be a
dvipng error! exitcode was 2 (signal 0), transscript follows:group with
dvipng error! exitcode was 2 (signal 0), transscript follows:infinite. Is it true that
dvipng error! exitcode was 2 (signal 0), transscript follows:admits a Dehn twist of infinite order?
Remark A Dehn twist of
dvipng error! exitcode was 2 (signal 0), transscript follows:is an automorphism
dvipng error! exitcode was 2 (signal 0), transscript follows:of the following form: either
dvipng error! exitcode was 2 (signal 0), transscript follows:splits as
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:is central,
dvipng error! exitcode was 2 (signal 0), transscript follows:for
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:for
dvipng error! exitcode was 2 (signal 0), transscript follows:; or
dvipng error! exitcode was 2 (signal 0), transscript follows:is an HNN extension
dvipng error! exitcode was 2 (signal 0), transscript follows:, with stable letter
dvipng error! exitcode was 2 (signal 0), transscript follows:,
dvipng error! exitcode was 2 (signal 0), transscript follows:is central,
dvipng error! exitcode was 2 (signal 0), transscript follows:for
dvipng error! exitcode was 2 (signal 0), transscript follows:and
dvipng error! exitcode was 2 (signal 0), transscript follows:.
It was shown by Levitt MR2174093 that a one ended hyperbolic group with
dvipng error! exitcode was 2 (signal 0), transscript follows:infinite admits a Dehn twist of infinite order coming from a splitting over a virtually cyclic group.