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

Abstract

Let ${\Pi }_g$ be the surface group of genus $g(g\ge 2)$, and denote by ${\mathcal{R}}_{{\Pi }_g}$ 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 ${\mathcal{R}}_{{\Pi }_g}$ formed by those $\phi$ which are semiconjugate to $k$-fold lifts of some homomorphisms and which have Euler number $eu(\phi )=l$ is shown to be clopen. This leads to a new proof of the main result of Kathryn Mann [Man] from a completely different approach.