show · curve.highergenus.aut.topological_equivalence all knowls · up · search:

Given a group $G$ acting on a compact Riemann surface $X$ by the maps $\phi_1: G \hookrightarrow \mathrm{Aut}(X)$ and $\phi_2: G \hookrightarrow \mathrm{Aut}(X)$, then $\phi_1$ and $\phi_2$ are topologically equivalent if there is an automorphism $\omega$ of $G$ and an orientation preserving homeomorphism $h: X \to X$ so that

$$\phi_2(g)=h \phi_1(\omega(g)) h^{-1} \text{ for all } g \in G.$$

Equivalently, if the actions are described as surjective homomorphisms $\eta_1: \Gamma \to G$ and $\eta_2: \Gamma \to G$ from a Fuchsian group $\Gamma$ with kernel isomorphic to the fundamental group of $X$, these actions are topologically equivalent if there is an automorphism $\omega$ of $G$ and a homeomorphism $h$ of the upper half plane $\mathbb{H}$ normalizing $\Gamma$ such that the following diagram commutes.

$$\begin{matrix}\Gamma & \xrightarrow{\mathit{\eta_1}}& G \\ \Big\downarrow h & & \Big\downarrow \omega \\ \Gamma & \xrightarrow{\mathit{\eta_2}}& G \end{matrix}$$

Authors:
Knowl status:
• Review status: beta
• Last edited by Jennifer Paulhus on 2020-07-16 18:51:25
History: