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 $\RR_{Π_g}$ 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 $\RR_{Π_g}$ 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 \cite{Mann} from a completely different approach.

25 pages, 11 figures

Cited by in corpus (1)

• Rigidity and flexibility of group actions on the circle