Hyperbolicity properties of quotient surfaces by freely operating arithmetic lattices
The authors are dealing with hyperbolicity properties of quotient surfaces. They prove the following theorem: Let \(D\) be a symmetric domain in \(\mathbb{C}^2\) and \(\Gamma\subset \Aut^0D\) an irreducible arithmetic lattice which operates freely on \(D\). Then the cusp-compactification \(\overline X\) of \(X= D/\Gamma\) is hyperbolic (cusp-compactification in the sense of Baily-Borel). As a corollary they obtain the following: The minimal resolution of the singularities of \(\overline X,\widehat X@>\pi>>\overline X\) is of general type and hyperbolic \(\text{mod }R\), \(R= \pi^{-1}(\overline X\setminus X)\), i.e., if \(d_{\widehat X}\) is the Kobayashi pseudometric on \(\widehat X\), \(d_{\widehat X}(x,y)= 0\) implies \(x= y\) or that \(x,y\in R\).
scientific article; zbMATH DE number 1409068
scientific article