A couple of real hyperbolic disc bundles over surfaces

Abstract

Applying the techniques developed in [AGG], we construct new real hyperbolic manifolds whose underlying topology is that of a disc bundle over a closed orientable surface. By the Gromov-Lawson-Thurston conjecture [GLT], such bundles $M\to S$ should satisfy the inequality $|eM/\chi S|\leqslant1$, where $eM$ stands for the Euler number of the bundle and $\chi S$, for the Euler characteristic of the surface. In this paper, we construct new examples that provide a maximal value of $|eM/\chi S|=\frac35$ among all known examples. The former maximum, belonging to Feng Luo [Luo], was $|eM/\chi S|=\frac12$.