Geodesic
The Automorphic Conjugacy Problem for Subgroups of Fundamental Groups of Compact Surfaces
Algebra i logika, Tome 40 (2001) no. 1, pp. 30-59.

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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.
Keywords: fundamental groups of compact surfaces, automorphic conjugacy problem for subgroups. 
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     author = {O. V. Bogopolskii},
     title = {The {Automorphic} {Conjugacy} {Problem} for {Subgroups} of {Fundamental} {Groups} of {Compact} {Surfaces}},
     journal = {Algebra i logika},
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     number = {1},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2001_40_1_a2/}
}
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O. V. Bogopolskii. The Automorphic Conjugacy Problem for Subgroups of Fundamental Groups of Compact Surfaces. Algebra i logika, Tome 40 (2001) no. 1, pp. 30-59. http://geodesic.mathdoc.fr/item/AL_2001_40_1_a2/