2023/06/27 by Jiming Ma, Ma, Jiming
Mathematics · #Advanced Algebra and Geometry #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2306.15240
openalex publication_date 2023/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We initiate the study of deformations of groups in three-dimensional complex hyperbolic geometry. Let G=⟨ ι1, ι2, ι3, ι4 | ι12= ι22 = ι32=ι42=id,
(ι1 ι3)2=(ι1 ι4)3=(ι2 ι4)2=id⟩ be an abstract group. We study representations ρ: G → PU(3,1), where ρ( ιi)=Ii is a complex reflection fixing a complex hyperbolic plane in \bf H3\mathbb C for 1 ≤ i ≤ 4, with the additional condition that I1I2 is parabolic. When we assume two pairs of hyper-parallel complex hyperbolic planes have the same distance, then the moduli space M is parameterized by (h,t) ∈ [1, ∞) × [0, π] but t ≤ arccos(-(3h2+1)/(4h2)). In particular, t=0 and t=arccos(-(3h2+1)/(4h2)) degenerate to \bf H3\mathbb R-geometry and \bf H2\mathbb C-geometry respectively. Using the Ford domain of ρ(√(2),arccos(-(7)/(8)))(G) as a guide, we show ρ(h,t) is a discrete and faithful representation of G → PU(3,1) when (h,t) ∈ M is near to (√(2), arccos(-(7)/(8))). This is the first nontrivial example of the Ford domain of a subgroup in PU(3,1) that has been studied.