Open Sets of Maximal Dimension in Complex Hyperbolic Quasi-Fuchsian Space

Abstract

Let π be the fundamental group of a closed surface Σ of genus g > 1. One of the fundamental problems in complex hyperbolic geometry is to find all discrete, faithful, geometrically finite and purely loxodromic representations of π into SU(2, 1), (the triple cover of) the group of holomorphic isometries of H²_C. In particular, given a discrete, faithful, geometrically finite and purely loxodromic representation ρ₀ of π₁, can we find an open neighbourhood of ρ₀ comprising representations with these properties. We show that this is indeed the case when ρ₀ preserves a totally real Lagrangian plane.