Equivalence of Group Actions on Riemann Surfaces

TL;DR

The paper constructs specific families of Riemann surfaces demonstrating that topological conjugacy of automorphism groups does not imply conformal conjugacy, affecting the structure of certain subvarieties in the moduli space.

Contribution

It introduces explicit examples of Riemann surfaces with automorphism groups that are topologically but not conformally conjugate, showing the non-normality of related subvarieties in moduli space.

Findings

Automorphism groups with conjugate cyclic subgroups are not necessarily conformally conjugate.

The subvariety al M_g(H_1) is not a normal subvariety in the moduli space.

Explicit families of Riemann surfaces illustrating these properties.

Abstract

We produce for each natural number $n \geq 3$ two 1--parameter families of Riemann surfaces admitting automorphism groups with two cyclic subgroups $H_{1}$ and $H_{2}$ of orden $2^{n}$, that are conjugate in the group of orientation--preserving homeomorphism of the corresponding Riemann surfaces, but not conjugate in the group of conformal automorphisms. This property implies that the subvariety $\mathcal{M}_{g}(H_{1})$ of the moduli space $\mathcal{M}_{g}$ consisting of the points representing the Riemann surfaces of genus $g$ admitting a group of automorphisms topologically conjugate to $H_{1} $ (equivalently to $H_{2}$) is not a normal subvariety.