2023/06/27 by Jiming Ma, Ma, Jiming
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2306.15707
openalex publication_date 2023/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let T be an infinite volume Coxeter tetrahedron in three dimensional real hyperbolic space \bf H3\mathbb R with two opposite right-angles and the other angles are all zeros. Let G be the Coxeter group of T, so G=⟨ ι1, ι2, ι3, ι4 | ι12= ι22 = ι32=ι42=id,
(ι1 ι3)2=(ι2 ι4)2=id⟩ as an abstract group. We study type-preserving representations ρ: G → PU(3,1), where ρ( ιi)=Ii is a complex reflection fixing a complex hyperbolic plane in three dimensional complex hyperbolic space \bf H3\mathbb C for 1 ≤ i ≤ 4. The moduli space M of these representations is parameterized by θ∈ [(5 π)/(6), π]. In particular, θ=(5 π)/(6) and θ=π degenerate to \bf H2\mathbb C-geometry and \bf H3\mathbb R-geometry respectively. Via Dirichlet domains, we show ρ=ρθ is a discrete and faithful representation of the group G for all θ∈ [(5 π)/(6), π]. This is the first nontrivial moduli space in three dimensional complex hyperbolic space that has been studied completely.