Twisted book decompositions and the Goeritz groups

TL;DR

This paper investigates the Goeritz groups associated with Heegaard splittings derived from twisted book decompositions, revealing cases with infinite-order mapping class groups not arising from open book decompositions, and provides explicit computations.

Contribution

It introduces a new class of Heegaard splittings with infinite-order Goeritz groups not induced by open book decompositions, with explicit group calculations.

Findings

Existence of distance 2 Heegaard splittings with infinite-order mapping class groups.

Identification of cases not arising from open book decompositions.

Explicit computation of the associated mapping class groups.

Abstract

We consider the Goeritz groups of the Heegaard splittings induced from twisted book decompositions. We show that there exist Heegaard splittings of distance $2$ that have the infinite-order mapping class groups whereas that are not induced from open book decompositions. Explicit computation of those mapping class groups are given.