Geodesic
Groups of signature $(0;n;0)$
Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 1, pp. 259-262.

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Let $M$ be an ideal polygon with $2n-2$ vertices. Consider a pairing of the symmetrical (with respect to some fixed diagonal) sides of $M$ by mappings $S_i$, $1\le i\le n-1$, and denote by $\Gamma$ the group generated by these mappings. Each $S_i$ depends on one parameter. We prove a necessary and sufficient condition for the possibility of choosing these parameters so that our polygon $M$ would be a fundamental domain for the action of $\Gamma$. 
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P. V. Tumarkin. Groups of signature $(0;n;0)$. Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 1, pp. 259-262. http://geodesic.mathdoc.fr/item/FPM_2003_9_1_a15/

[1] Puankare A., Izbrannye trudy, T. 3, Nauka, M., 1971

[2] Berdon A., Geometriya diskretnykh grupp, Nauka, M., 1986 | MR