A construction of representations of \(3\)-manifold groups into \(\mathrm{PU}(2,1)\) through Lefschetz fibrations (Q6630430)

The main objective of this paper is to introduce a new method for constructing infinitely many non-conjugative representations of the fundamental groups of the closed hyperbolic \(3\)-dimensional manifolds into a lattice in the holomorphic isometry group \(\mathrm{Isom}(\mathbb H^2_{\mathbb C})\) of the complex hyperbolic plane. The representations do not arise as holonomies of uniformizable structures since the domains of their discontinuities are empty. The primary approach toward constructing the already mentioned representations relies upon a careful examination of the properties of a complex hyperbolic surface. In particular, it concentrates on the specific example of Hirzebruch's surface.