A couple of real hyperbolic disc bundles over surfaces

TL;DR

This paper constructs new real hyperbolic disc bundles over surfaces, achieving a higher Euler number to Euler characteristic ratio than previously known, thus providing new insights into the topology of hyperbolic manifolds.

Contribution

It introduces new examples of hyperbolic disc bundles with a maximal Euler number to Euler characteristic ratio of 3/5, surpassing previous known examples.

Findings

Achieved a ratio of |eM/χS|=3/5 in hyperbolic disc bundles

Constructed explicit examples of hyperbolic manifolds with new topological properties

Extended the known range of Euler number to Euler characteristic ratios in hyperbolic geometry

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 $|eM/\chi S|=\frac12$.