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

Abstract

Let \(M=H_1\cup _S H_2\) be a Heegaard splitting of a closed orientable 3-manifold M (or a bridge decomposition of a link exterior). Consider the subgroup \({\text {MCG}}^0(H_j)\) 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 \(\mathcal {CC}(S)\) obtained as the image of \({\text {MCG}}^0(H_j)\). Then the group \(G=\langle G_1, G_2\rangle \) generated by \(G_1\) and \(G_2\) acts on \(\mathcal {CC}(S)\) 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.