A couple of real hyperbolic disc bundles over surfaces

Abstract

Applying the techniques developed in \[1], 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 \[6], such bundles $M \to S$ should satisfy the inequality $|eM/\chi S|\leq 1$, 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 $|eM/\chi S|=\frac {3}{5}$ among all known examples. The former maximum, belonging to Feng Luo \[10], was $|eM/\chi S|=\frac {1}{2}$.