Genus two Goeritz groups of lens spaces

TL;DR

This paper proves that the Goeritz groups for genus-2 Heegaard splittings of lens spaces $L(p, 1)$ are finitely presented and provides explicit group presentations.

Contribution

It establishes finite presentability of these groups and explicitly describes their algebraic structure.

Findings

Goeritz groups are finitely presented for lens spaces $L(p, 1)$

Explicit presentations of these groups are provided

Enhances understanding of symmetries in 3-manifold topology

Abstract

Given a genus-$g$ Heegaard splitting of a 3-manifold, the Goeritz group is defined to be the group of isotopy classes of orientation-preserving homeomorphisms of the manifold that preserve the splitting. In this work, we show that the Goeritz groups of genus-2 Heegaard splittings for lens spaces $L(p, 1)$ are finitely presented, and give explicit presentations of them.