A note on large bounding and non-bounding finite group-actions on surfaces of small genus

TL;DR

This paper investigates which finite group-actions on small genus surfaces are bounding or geometrically bounding, focusing on large group-actions on genus 3 surfaces, and clarifies their extension properties to 3-manifolds.

Contribution

It provides a classification of bounding and geometrically bounding actions for large groups on genus 3 surfaces, extending known results to new cases.

Findings

Identifies which group-actions extend to handlebodies

Determines conditions for actions to extend to hyperbolic 3-manifolds

Focuses on large group-actions on genus 3 surfaces

Abstract

The classification of finite group-actions on closed surfaces of small genus is well-known. In the present paper we are interested in the question of which of these group-actions are bounding (extend to a compact 3-manifold with the surface as its unique boundary component, e.g. to a handlebody) or geometrically bounding (extend to a hyperbolic 3-manifold with totally geodesic boundary), concentrating, as a typical case, on large group-actions on surfaces of genus 3.