Let $S$ be a closed oriented surface of genus $g>1$. Is the following true ?
Let $\alpha,\beta\in \pi_1(S)\backslash \{1\}$ and $\rho:\pi_1(S)\rightarrow PSL(2,\mathbb R)$ be a discrete faithful representation. Assume that there exists a horocycle $h$ in $\mathbb H^2$ such that the horocycles $h, \rho(\alpha)(h)$ and $\rho(\beta)(h)$ have the same based point. Then $\alpha=\beta$.
Here, the based point of a horocycle $h$ in $\mathbb H$ is the unique tangent point of $h$ to the boundary of $\mathbb H$.
In the first place, you are asking that $h$ and $\rho(a)h$ have the same base point, but this means $\rho(a)$ will fix that point on the boundary. Then the entire orbit $\rho(a^n)$ will fix the point, so any $b=a^n$ will satisfy your condition with $b\neq a$.