I am self-studying hyperbolic geometry and am stuck on the following.
Let $\mathbb{D}$ be the Poincare disc model of the hyperbolic plane with $[3^7]$ tiling on it. Let $T$ be a hyperbolic translation which is also a symmetry of the tiling such that $T(p)=q$ where $p$ and $q$ are two vertices. Then is this $T$ unique?
Since $T$ is a translation it has an axis of translation. Is it possible that some other translation has the same axis and maps $p$ to $q$?
Thanks in advance.