Translation surfaces of finite type in ${\rm Sol}_{3}$

Abstract

In the homogeneous space Sol$_{3}$, a translation surface is parametrized by $r(s,t)=\gamma _{1}(s)\ast \gamma _{2}(t)$, where $\gamma _{1}$ and $\gamma _{2}$ are curves contained in coordinate planes. In this article, we study translation invariant surfaces in ${\rm Sol}_{3}$, which has finite type immersion.