Complex hyperbolic $(3,3,n)$ triangle groups

John R. Parker, Jieyan Wang and Baohua Xie

Vol. 280 (2016), No. 2, 433–453

DOI: 10.2140/pjm.2016.280.433

Abstract

Let $p,q,r$ be positive integers. Complex hyperbolic $\left(p,q,r\right)$ triangle groups are representations of the hyperbolic $\left(p,q,r\right)$ reflection triangle group to the holomorphic isometry group of complex hyperbolic space $H_{ℂ}^2$, where the generators fix complex lines. In this paper, we obtain all the discrete and faithful complex hyperbolic $\left(3,3,n\right)$ triangle groups for $n\ge 4$. Our result solves a conjecture of Schwartz in the case when $p=q=3$.

Keywords

complex hyperbolic geometry, complex hyperbolic triangle groups

Mathematical Subject Classification 2010

Primary: 20H10

Secondary: 22E40, 51M10

Milestones

Received: 27 January 2015

Revised: 18 June 2015

Accepted: 2 July 2015

Published: 28 January 2016