Orbifolds and orbibundles in complex hyperbolic geometry

TL;DR

This paper develops a geometrical framework for orbibundles in complex hyperbolic geometry using diffeology, aiming to compute invariants like Euler numbers and Toledo invariants for orbibundles over 2-orbifolds.

Contribution

It introduces new tools for calculating invariants of complex hyperbolic disc orbibundles over 2-orbifolds, enhancing understanding in 4-manifold geometry.

Findings

New methods for computing Euler numbers of orbibundles

Calculation techniques for Toledo invariants of group representations

Application of diffeology to complex hyperbolic orbifold theory

Abstract

We develop the theory of orbibundles from a geometrical viewpoint using diffeology. One of our goals is to present new tools allowing to calculate invariants of complex hyperbolic disc orbibundles over $2$-orbifolds appearing in the geometry of $4$-manifolds. These invariants are the Euler number of disc orbibundles and the Toledo invariant of $\mathrm{PU}(2,1)$-representations of $2$-orbifold groups.