Actions of groups of homeomorphisms on one-manifolds

TL;DR

This paper classifies all group homomorphisms from the group of compactly-supported homeomorphisms isotopic to the identity on a manifold to the homeomorphism groups of the real line or circle, revealing the structure of such actions.

Contribution

It provides a complete description of all possible actions of these homeomorphism groups on one-dimensional manifolds, a significant step in understanding their algebraic and dynamical properties.

Findings

All group morphisms are classified explicitly.

Actions on the real line are characterized completely.

Actions on the circle are fully described.

Abstract

In this article, we describe all the group morphisms from the group of compactly-supported homeomorphisms isotopic to the identity of a manifold to the group of homeomorphisms of the real line or of the circle.