The (4,p)-arithmetic hyperbolic lattices, p≥ 2, in three dimensions

2022/06/28 by G. J. Martin, Martin, G. J., K. Salehi +3
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2206.14174

openalex publication_date 2022/06/28 · openalex created_date 2022/07/01 · openalex updated_date 2026/07/28

Abstract

We identify the finitely many arithmetic lattices Γ in the orientation preserving isometry group of hyperbolic 3-space ℍ³ generated by an element of order 4 and and element of order p≥ 2. Thus Γ has a presentation of the form Γ≅⟨ f,g: f⁴=g^p=w(f,g)=⋯=1 ⟩ We find that necessarily p∈ \2,3,4,5,6,∞\, where p=∞ denotes that g is a parabolic element, the total degree of the invariant trace field kΓ=ℚ(\\tr²(h):h∈Γ\) is at most 4, and each orbifold is either a two bridge link of slope r/s surgered with (4,0), (p,0) Dehn surgery (possibly a two bridge knot if p=4) or a Heckoid group with slope r/s and w(f,g)=(w_r/s)^r with r∈ \1,2,3,4\. We give a discrete and faithful representation in PSL(2,ℂ) for each group and identify the associated number theoretic data.

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