The Automorphic Conjugacy Problem for Subgroups of Fundamental Groups of Compact Surfaces

Let $\Sigma$ be a compact connected surface with basepoint $x$ and $H_1$ and $H_2$ be two finitely generated subgroups of $\pi_1(\Sigma, x)$ on finite sets of generators. It is proved that there exists an algorithm which decides whether there is an automorphism $\alpha\in\operatorname{Aut}(\pi_1(\Sigma, x))$ for which $\alpha (H_1)=H_2$, and if so, it finds such.