Menger curve and Spherical CR uniformization of a closed hyperbolic 3-orbifold

TL;DR

This paper demonstrates that a specific hyperbolic group with a Menger curve boundary can be represented in PU(2,1) such that its associated 3-orbifold at infinity is a closed hyperbolic 3-orbifold with a particular singular locus, confirming part of Kapovich's conjecture.

Contribution

It constructs a faithful, discrete representation of a hyperbolic group into PU(2,1) leading to a closed hyperbolic 3-orbifold at infinity, confirming a conjecture by Kapovich.

Findings

The 3-orbifold at infinity is a closed hyperbolic 3-orbifold.

The underlying space of the orbifold is the 3-sphere.

The singular locus is a Z_3-coned chain-link C(6,-2).

Abstract

Let $$G_{6,3}=\langle a_0, \cdots, a_5| a_{i}^{3}=id, a_{i} a_{i+1}= a_{i+1} a_{i}, i \in \mathbb{Z}/6\mathbb{Z}\rangle$$ be a hyperbolic group with boundary the Menger curve. J. Granier \cite{Granier} constructed a discrete, convex cocompact and faithful representation $\rho$ of $G_{6,3}$ into $\mathbf{PU}(2,1)$. We show the 3-orbifold at infinity of $\rho(G_{6,3})$ is a closed hyperbolic 3-orbifold, with underlying space the 3-sphere and singular locus the $\mathbb{Z}_3$-coned chain-link $C(6,-2)$. This answers the second part of Misha Kapovich's Conjecture 10.6\cite{Kapovich}.