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.
Categories: math.AG