Complex hyperbolic triangle groups

TL;DR

This paper discusses recent advances in complex hyperbolic triangle groups, focusing on their deformations, and reports the discovery of a real hyperbolic 3-manifold related to these groups.

Contribution

It introduces new findings on complex hyperbolic deformations of triangle groups and identifies a real hyperbolic 3-manifold at infinity for such groups.

Findings

Discovery of a closed real hyperbolic 3-manifold at infinity.

Progress in understanding complex hyperbolic deformations of triangle groups.

Insights into the structure of complex hyperbolic discrete groups.

Abstract

The theory of complex hyperbolic discrete groups is still in its childhood but promises to grow into a rich subfield of geometry. In this paper I will discuss some recent progress that has been made on complex hyperbolic deformations of the modular group and, more generally, triangle groups. These are some of the simplest nontrivial complex hyperbolic discrete groups. In particular, I will talk about my recent discovery of a closed real hyperbolic 3-manifold which appears as the manifold at infinity for a complex hyperbolic discrete group.