If a Riemann surface $S$ has genus $g\geq 2$, its automotphisms group is finite. I was wondering if there exists a hyperelliptic Riemann surface $S$ with $\# Aut(S)=2$. In other words, I was wondering if the hyperelliptic involution $J:S\rightarrow S$ is the generator of the automorphisms group of any hyperelliptic Riemann surface $S$.
Any help would be appreciated.
According to the results contained in
Poonen, Varieties without extra automorphisms II: hyperelliptic curves
for any genus $g\geq 2$ there exists a hyperelliptic Riemann surface with just two automorphisms: the identity and the hyperelliptic involution. On the other hand it is proved in
Gutierrez, Shaska, Hyperelliptic Curves with Extra Involutions
that the locus of genus $g$ hyperelliptic curves which admit extra involutions (besides the hyperelliptic one) is a $g$-dimensional subvariety of the moduli space of hyperelliptic curves.