On the interplay between discrete invariants of complex hyperbolic disc bundles over surfaces

We investigate the relationship between three natural invariants of complex hyperbolic disc orbibundles over oriented and closed hyperbolic $2$-orbifolds. These invariants are the Euler characteristic $\chi$ of the $2$-orbifold, the Euler number $e$ of the disc orbibundle, and the Toledo invariant $\tau$ of a faithful representation of the surface group into $\mathrm{P}\mathrm{U}(2,1)$ attached to the complex hyperbolic structure of the disc orbibundle. Based on previous examples, we conjecture that $-3\mathrm{\mid }\tau \mathrm{\mid }=2e+2\chi$ always holds. For complex hyperbolic disc orbibundles over $2$-orbifolds derived from quadrangles of bisectors via tessellation, we prove that $3\tau =2e+2\chi$. Furthermore, we demonstrate that $-3\mathrm{\mid }\tau \mathrm{\mid }=2e+2\chi$ holds when a section with no complex tangent planes is present.