A construction of representations of 3-manifold groups into PU(2,1) through Lefschetz fibrations

TL;DR

This paper constructs numerous non-conjugate representations of 3-manifold groups into PU(2,1) using Lefschetz fibrations and complex hyperbolic orbifolds, expanding understanding of 3-manifold symmetries.

Contribution

It introduces a method to produce infinitely many distinct representations of 3-manifold groups into PU(2,1) via Lefschetz fibrations and hyperbolic orbifolds.

Findings

Infinitely many non-conjugate representations constructed

Representations relate 3-manifolds to complex hyperbolic orbifolds

Uses branched coverings and hyperplane arrangements

Abstract

We obtain infinitely many (non-conjugate) representations of 3-manifold fundamental groups into a lattice in the holomorphic isometry group of complex hyperbolic space. The lattice is an orbifold fundamental group of a branched covering of the projective plane along an arrangement of hyperplanes constructed by Hirzebruch. The 3-manifolds are related to a Lefschetz fibration of the complex hyperbolic orbifold.