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

Abstract

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 $\mathrm{PU}(2,1)$ attached to the complex hyperbolic structure of the disc orbibundle. Based on previous examples, we conjecture that $-3|\tau |=2e+2\chi$ 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 $-3|\tau |=2e+2\chi$ holds when a section with no complex tangent planes is present.

13 pages, 3 figures