Subgroups of mapping class groups related to Heegaard splittings and bridge decompositions

Let M = H 1 ∪ S H 2 be a Heegaard splitting of a closed orientable 3-manifold M (or a bridge decomposition of a link exterior). Consider the subgroup MCG 0 ( H j ) of the mapping class group of H j consisting of mapping classes represented by orientation-preserving auto-homeomorphisms of H j homotopic to the identity, and let G j be the subgroup of the automorphism group of the curve complex CC ( S ) obtained as the image of MCG 0 ( H j ) . Then the group G = ⟨ G 1 , G 2 ⟩ generated by G 1 and G 2 acts on CC ( S ) with each orbit being contained in a homotopy class in M . In this paper, we study the structure of the group G and examine whether a homotopy class can contain more than one orbit. We also show that the action of G on the projective lamination space of S has a non-empty domain of discontinuity when the Heegaard splitting satisfies R -bounded combinatorics and has high Hempel distance.