Geodesic
Hyperbolic Chevalley Groups on $\mathbb{C}^2$
Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 64-72

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Let $\Gamma\subset U(1,1)$ be the subgroup generated by the complex reflections. Suppose that $\Gamma$ acts discretely on the domain $K=\{(z_1,z_2)\in\mathbb{C}^2\mid |z_1|^2-|z_2|^20\}$ 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 $\mathbb{C}^2-\{0\}$.
Keywords: reflection group, Fuchsian group, Chevalley theorem.
Mots-clés : quotient space 
@article{FAA_2009_43_2_a4,
     author = {O. V. Schwarzman},
     title = {Hyperbolic {Chevalley} {Groups} on $\mathbb{C}^2$},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {64--72},
     publisher = {mathdoc},
     volume = {43},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a4/}
}
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O. V. Schwarzman. Hyperbolic Chevalley Groups on $\mathbb{C}^2$. Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 64-72. http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a4/