Basic partitions and combinations of group actions on the circle: A new approach to a theorem of Kathryn Mann

TL;DR

This paper introduces a novel approach using basic partitions and group action combinations to analyze surface group representations on the circle, providing a new proof of Kathryn Mann's theorem.

Contribution

It presents a new method based on partitions and group actions to reprove a key theorem in surface group representations on the circle.

Findings

The subset of representations with specific lifts and Euler number is clopen.

A new proof of Kathryn Mann's main result is established.

The approach offers a different perspective on surface group actions.

Abstract

Let $\Pi_g$ be the surface group of genus $g$ ($g\geq2$), and denote by $\RR_{\Pi_g}$ the space of the homomorphisms from $\Pi_g$ into the group of the orientation preserving homeomorphisms of $S^1$. Let $2g-2=kl$ for some positive integers $k$ and $l$. Then the subset of $\RR_{\Pi_g}$ formed by those $\varphi$ which are semiconjugate to $k$-fold lifts of some homomorphisms and which have Euler number $eu(\varphi)=l$ is shown to be clopen. This leads to a new proof of the main result of Kathryn Mann \cite{Mann} from a completely different approach.