The measure of the limit set of the handlebody group (Q749929)

The homeotopy group H of a handlebody \(M_ g\) of genus g can be viewed as a subgroup of the homeotopy group of the surface \(F_ g=\partial M_ g\). There exists a natural action of H on the space \({\mathcal P}{\mathcal F}\) of projective classes of measured foliations on \(F_ g\). The author proves that the limit set \(\Lambda\) of the action has measure zero. The same result holds for any orientable irreducible 3-manifold \(M^ 3\) with the surface \(F_ g\) of genus \(g>1\) as a boundary component except for the case when \(M=F_ g\times [0,1]\). In the proof the train-track language and H. Mazur's characterization of the limit set are used.