Hilden Braid Groups

TL;DR

This paper explores generalized Hilden groups within the $2n$-punctured mapping class group of a genus g surface, linking them to Heegaard splittings, bridge decompositions, and link motion groups in 3-manifolds.

Contribution

It introduces new subgroups generalizing Hilden groups and establishes their connections to manifold decompositions and link motion groups.

Findings

Generalized Hilden groups relate to Heegaard splittings.

Connections established with bridge decompositions of links.

Links to motion groups of links in closed 3-manifolds.

Abstract

Let $\H_g$ be a genus $g$ handlebody and $\MCG_{2n}(\T_g)$ be the $2n$-punctured mapping class group of $\T_g=\partial\H_g$. In this paper we study two particular subgroups of $\MCG_{2n}(\T_g)$ which generalize Hilden groups. As well as Hilden groups are related to plate closures of braids, these generalizations are related to Heegaard splittings of manifolds and to bridge decompositions of links. Connections between these subgroups and motion groups of links in closed 3-manifolds are also provided.