Recall that a JSJ decomposition for a 3-manifold is a (minimal) embedded collection of tori that cut the 3-manifold into pieces such that each component is atoroidal or Seifert fibered. Jaco-Shalen and Johannson (i.e. JSJ) showed that up to isotopy, every irreducible 3-manifold has exactly one such decomposition (which is empty, if the 3-manifold is already atoroidal or Seifert fibered). If you’re also interested in Heegaard splittings of the 3-manifold in question (which I usually am) then you can consider a result of Tsuyoshi Kobayashi, that a strongly irreducible Heegaard splitting can always be isotoped to intersect a given incompressible torus in simple closed curves that are essential in both surface.

One nice application of this result, also due to Kobayashi, is a classification [1] of 3-manifolds with genus two Heegaard splittings and non-trivial JSJ decompositions. In a recent preprint [2], Jungsoo Kim classifies finite group actions on these three manifolds that preserve both the Heegaard surface and the JSJ tori. He considers each of three cases identified by Kobayashi and shows that in each case, the only possible (non-trivial) group actions are by the order two cyclic group or the order four dihedral group.

These group actions are closely related to the mapping class groups of the Heegaard splittings (the group of automorphisms of the 3-manifold that take the Heegaard surface onto itself, modulo isotopies that preserve the Heegaard surface.) In particular, the automorphisms of the 3-manifold induce a subgroup of the mapping class group of the Heegaard splitting. In this case, I think the induced subgroup has to be isomorphic to the acting group, since a finte group cannot act on a genus two surface by isotopy trivial automorphisms. (I don’t have a reference for this.)

What I’d like to know is: Is the whole mapping class group of the Heegaard splitting isomorphic to a maximal group acting on the 3-manifold in a way that preserves both the Heegaard surface and the JSJ surface. In other words, can the mapping class group be realized by on of Kim’s group actions. Hyam Rubinstein and I proved in [3] that every finite subgroup of the mapping class group of a Heegaard splitting can be realized by a group of automorphisms (essentially, a group action). The problem is that an automorphism that preserves the Heegaard splitting may not preserve the JSJ tori. Such an automorphism will take the JSJ surface to an isotopic surface, but this surface may intersect the Heegaard surface differently, so that there is not an isotopy preserving the Heegaard surface that takes the new JSJ tori to the originals. At least, this is the problem when considering JSJ tori in general. In this case, because the ways the JSJ tori can intersect a genus two surface are limited, and mostly understood, it seems like there might be a chance of showing that there’s only one way (up to isotopy) for the JSJ tori to intersect the Heegaard surface. But I don’t understand Kobayashi’s classification well enough to figure it out, so for now it’s open.

[1] T. Kobayashi, Structures of the Haken manifolds with Heegaard splittings of genus two, Osaka J. Math. 21 (1984), 437{455.

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