Let $C$ be a (very) general genus 1 curve embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a (2,2)-divisor. 


Each projection defines $C$ as a double cover of $\mathbb{CP}^1$ and induces an involution $\tau\_i:C\to C$. Let $G\simeq \mathbb Z/2 \* \mathbb Z/2$ be the subgroup of $Aut(C)$ generated by these. Note that $G$ is infinite.


Are there points on $C$ with finite orbit under $G$?



Finite orbits on an elliptic curve with two generic involutions
No. Letting $\sigma$ and $\tau$ denote the two involutions, $\sigma \circ \tau$ is a translation by an element of $\mathrm{Pic}^0(C)$. In general, this translation will not be torsion, so its orbit through any point is infinite. (In fact, if the translation IS torsion, then $G$ is a finite dihedral group, not $\mathbb{Z}/2 \ast \mathbb{Z}/2$.)


