Equivalence of Group Actions on Riemann Surfaces

Details Equivalence of Group Actions on Riemann Surfaces Equivalence of Group Actions on Riemann Surfaces

2010-12-20

arxiv.org/abs/1012.4337v1

Mathematics

Algebraic Geometry

20 pages, 2 figures

Scientific paper

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.

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