Juan Souto has now shown that the action of the mapping class group on the unit tangent bundle of a surface can’t be “smoothed” to act by diffeomorphisms.

The only caveat is that the genus of the surface must be >11, and one must take the full mapping class group, including the orientation reversing maps.

http://www-personal.umich.edu/~jsouto/papers.html

As you remarked, such a generalization does not seem to be useful in the study of hyperbolic structures on manifolds but rather has something to do with Mobius structures on manifolds. See, for example, Kulkarni and Pinkall, Propositions (3.3.ii) and (4.6):

]]>However, I’m not sure what applications it would have. If M^3 is a finite volume hyperbolic 3-manifold, then one can deduce from Mostow rigidity that the mapping class group of M^3 is exactly the isometry group of M^3. In particular, it is a finite group and it acts on M^3 (no need to pass to the unit tangent bundle). For an infinite volume hyperbolic 3-manifold, one can still lift any diffeomorphism to the universal cover and get an action on S^3. However, that action need not preserve round circles (if it always did, then one could extend Mostow rigidity to infinite volume 3-manifolds, which is wildly false). Thus we would not get an action on our parameterization of the unit tangent bundle.

]]>Wouldn’t it better if v is “in the direction of K” so that this construction generalizes to higher dimensions?

]]>Take triples of points on this circle, which may be identified with the space of ideal triangles in the hyperbolic plane, and therefore with the unit tangle bundle. Each mapping class element lifts to a homeomorphism of the universal cover of the surface, and therefore of the triple points of the circle at infinity. This lift is only well-defined up to covering translations, so when we quotient the space of triple points by the covering translations, we get a well-defined canonical action on the unit tangent bundle to the surface.

Can this action be realized by diffeomorphisms? I suspect the answer is no, given the nature of the action of each element on the circle at infinity.

For the Heegaard splitters, can this action preserve a Heegaard splitting?