A complex hyperbolic Riley slice

Parker, John¹ ; Will, Pierre²

¹ Department of Mathematical Sciences, Durham University, Durham, United Kingdom
² Université Grenoble Alpes, Institut Fourier, Saint-Martin-d’Hères, France

Geometry & topology, Tome 21 (2017) no. 6, pp. 3391-3451.

Résumé

We study subgroups of $PU\left(2,1\right)$ generated by two noncommuting unipotent maps $A$ and $B$ whose product $AB$ is also unipotent. We call $\mathcal{U}$ the set of conjugacy classes of such groups. We provide a set of coordinates on $\mathcal{U}$ that make it homeomorphic to ${ℝ}^2$. By considering the action on complex hyperbolic space $H_{ℂ}^2$ of groups in $\mathcal{U}$, we describe a two-dimensional disc $\mathcal{Z}$ in $\mathcal{U}$ that parametrises a family of discrete groups. As a corollary, we give a proof of a conjecture of Schwartz for $\left(3,3,\infty \right)$–triangle groups. We also consider a particular group on the boundary of the disc $\mathcal{Z}$ where the commutator $\left[A,B\right]$ is also unipotent. We show that the boundary of the quotient orbifold associated to the latter group gives a spherical CR uniformisation of the Whitehead link complement.

Détail

DOI : 10.2140/gt.2017.21.3391

Classification : 20H10, 22E40, 51M10, 57M50
Keywords: discrete subgroups of Lie groups, complex hyperbolic geometry, spherical CR structures, complex hyperbolic quasi-Fuchsian groups

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