Actions of the group of homeomorphisms of the circle on surfaces

TL;DR

This paper classifies all group homomorphisms from the circle's orientation-preserving homeomorphisms to those of the annulus and torus, providing a comprehensive understanding of their actions on these surfaces.

Contribution

It explicitly describes all such group morphisms, a novel classification in the context of surface homeomorphism groups.

Findings

Complete classification of group morphisms to annulus and torus homeomorphisms

Identification of structural properties of these group actions

Insights into the dynamics of circle homeomorphism groups on surfaces

Abstract

In this article, we describe all the group morphisms from the group of orientation-preserving homeomorphisms of the circle to the group of homeomorphisms of the annulus or of the torus.