A uniformizable spherical CR structure on a two-cusped hyperbolic 3–manifold

Jiang, Yueping; Wang, Jieyan; Xie, Baohua

doi:10.2140/agt.2023.23.4143

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A uniformizable spherical CR structure on a two-cusped hyperbolic $3$–manifold

Yueping Jiang, Jieyan Wang and Baohua Xie

Algebraic & Geometric Topology 23 (2023) 4143–4184

DOI: 10.2140/agt.2023.23.4143

Abstract

Let $⟨ I_{1}, I_{2}, I_{3} ⟩$ be the complex hyperbolic $(4, 4, \infty)$ triangle group. We prove Schwartz’s conjecture that $⟨ I_{1}, I_{2}, I_{3} ⟩$ is discrete and faithful if and only if $I_{1} I_{3} I_{2} I_{3}$ is nonelliptic. If $I_{1} I_{3} I_{2} I_{3}$ is parabolic, we show that the even subgroup $⟨ I_{2} I_{3}, I_{2} I_{1} ⟩$ is the holonomy representation of a uniformizable spherical CR structure on the two-cusped hyperbolic $3$ –manifold $s 782$ in SnapPy notation.

Keywords

complex hyperbolic space, spherical CR uniformization, triangle groups, Ford domain, hyperbolic $3$–manifolds

Mathematical Subject Classification

Primary: 20H10, 22E40, 51M10, 57M50

References

Publication

Received: 19 April 2021

Revised: 27 February 2022

Accepted: 23 June 2022

Published: 23 November 2023

Authors

Yueping Jiang
School of Mathematics Hunan University Changsha China

Jieyan Wang
School of Mathematics Hunan University Changsha China

Baohua Xie
School of Mathematics Hunan University Changsha China

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Volume 23, issue 9 (2023)