Hyperbolic Chevalley Groups on $\mathbb{C}^2$

Let $\Gamma\subset U(1,1)$ be the subgroup generated by the complex reflections. Suppose that $\Gamma$ acts discretely on the domain $K=\{(z_1,z_2)\in\mathbb{C}^2\mid |z_1|^2-|z_2|^2<0\}$ and that the projective group $P\Gamma$ acts on the unit disk $B=\{|z_1/z_2|<1\}$ as a Fuchsian group of signature $(n_1,\dots,n_s)$, $s\ge 3$, $n_i\ge 2$. For such groups, we prove a Chevalley type theorem, i.e., find a necessary and sufficient condition for the quotient space $K/\Gamma$ to be isomorphic to $\mathbb{C}^2-\{0\}$.