Sufficient conditions for a group of automorphisms of a Riemann surface to be its full automorphism group

A necessary and sufficient condition for lifting the hyperelliptic involution

“…From Theorem III.4, there exists a Riemann surface X such that X/H = Y and the involution of Y does not lift to an automorphism of X. If g = 3, this result, in conjunction with a theorem of [12], yields that H = Aut(X). We outline the analogous argument if g = 2.…”

“…But in this case, we apply Proposition IV.3, which yields that x 2g+1 + 2H = x 1 + x 2 + · · · + x 2g + 2H. Thus, in this case also, H equals the union in (12), and there is no fixed point free lift. b) We now show that if 2 r−s > 2, then there exists a Riemann surface which admits H as a group of automorphisms which has a fixed point free lift of the hyperelliptic involution.…”

“…To accomplish this, we employ techniques developed in [12] to deal with covering surfaces of (not necessarily hyperelliptic) Riemann surfaces. In [12], a generalization of the following theorem is proved. Proof.…”

“…A complete determination is made of the abelian groups which may arise as automorphism groups of surfaces which possess a fixed point free lift. These results are combined with results in [12] to yield specific examples of automorphism groups yielding a hyperelliptic orbit space.…”

On extendability of group actions on compact Riemann surfaces

“…For example, the case of surfaces of low genus was carefully studied by Wiman [14] in the late 1800s. Wiman directly handled defining equations of the surfaces, as used also in [5], [12] and [13]. Similarly, the technique of Fuchsian groups was used in [4] to solve the problem for the family of hyperelliptic Riemann surfaces.…”