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

TL;DR

This paper investigates the structure of subgroups of the mapping class group related to Heegaard splittings and bridge decompositions, focusing on how these subgroups preserve simple loops and exploring the extent of the converse property.

Contribution

It introduces a detailed analysis of the subgroup structure of mapping class groups associated with Heegaard splittings and bridge decompositions, and examines conditions under which simple loops are preserved.

Findings

The group G preserves the homotopy class of simple loops on the splitting surface.

Conditions under which the subgroup G characterizes the preservation of simple loops.

Insights into the structure of subgroups generated by automorphisms of the curve complex.

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.