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 ${\mathrm{MCG}}^0(H_j)$ of the mapping class group of $H_j$ consisting of mapping classes represented by 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 ${\mathrm{MCG}}^0(H_j)$. Then the group $G=<G_1,G_2>$ generated by $G_1$ and $G_2$ preserve the homotopy class in $M$ of simple loops on $S$. In this paper, we study the structure of the group $G$ and the problem to what extent the converse to this observation holds.

19 pages: the second version. The assumption of Theorem 3 has been changed from the bounded geometry to the bounded combinatorics

References in corpus (1)

• Epimorphisms between 2-bridge link groups