Groups of signature $(0;n;0)$

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 $\Gamma$.