The subgroup of the Goeritz group of the Heegaard splitting induced by an openbook decomposition consisting of elements preserving the binding

TL;DR

This paper investigates a specific subgroup of the Goeritz group associated with Heegaard splittings derived from openbook decompositions, focusing on elements that preserve the binding and exploring their algebraic structure and examples.

Contribution

It characterizes the subgroup of the Goeritz group preserving the binding in terms of monodromy and Dehn twists, providing a new algebraic description and criteria for certain symmetries.

Findings

The subgroup is the quotient of the monodromy-commuting subgroup by boundary Dehn twists.

A criterion for elements fixing the binding and reversing orientation is established.

Examples of computing the Goeritz group are provided.

Abstract

When a 3-manifold admits an openbook decomposition, we get a Heegaard splitting by thickening a page. This splitting surface has a special multi curves coming from the binding. In this paper, we consider the subgroup of the Goeritz group of this Heegaard splitting, which is the mapping class group of the 3-manifold preserving the given Heegaard splitting, consisting of elements preserving the binding. This subgroup turned out to be the quotient of the subgroup of the orientation preserving mapping class group consisting of elements commuting with the monodromy by the subgroup generated by the Dehn twists along the boundary curves. We also get a criterion for the existence of an element of the Goeritz group which fixes the binding as a set and reverses the orientation. At last, we give some example of computation of a Goeritz group.