In complex hyperbolic geometry we consider an open set biholomorphic to an open ball in Cn, and we equip it with a particular metric that makes it have constant negative holomorphic curvature. This is analogous to but different from the real hyperbolic space. In the complex case, the sectional curvature is constant on complex lines, but it changes when we consider real 2-planes which are not complex lines.

Complex Hyperbolic Geometry

We describe a general geometrical construction of representations of fundamental groups of 3-manifolds into PU(2, 1) and eventually of spherical CR structures defined on those 3-manifolds. We construct branched spherical CR structures on the complement of the figure eight knot and the Whitehead link. They have discrete holonomies contained in PU(2, 1,Z[ω]) and PU(2, 1,Z[i]) respectively.

/pdf/a-spherical-cr-structure-on-the-complement-of-the-figure-nusn1hik9i.pdf

A spherical CR structure on the complement of the figure eight knot with discrete holonomy

http://maths.dur.ac.uk/~dma0jrp/img/FenchelNielsen.pdf

Complex hyperbolic Fenchel–Nielsen coordinates

Jorgensen's picture of quasifuchsian punctured torus groups.- Fricke surfaces and PSL(2, ?)-representations.- Labeled representations and associated complexes.- Chain rule and side parameter.- Special examples.- Reformulation of Main Theorem 1.3.5 and outline of the proof.- Openness.- Closedness.- Algebraic roots and geometric roots.

Punctured torus groups and 2-bridge knot groups (I)

Geometric structures associated to triangulations as fixed point sets of involutions

In this work, we investigate how to decompose a pair (A, B) of loxodromic isometries of the complex hyperbolic plane H 2 under the form A = I1I2 and B = I3I2, where the Ik's are involutions. The main result is a decomposability criterion, which is expressed in terms of traces of elements of the group hA, Bi.

/pdf/traces-cross-ratios-and-2-generator-subgroups-of-su-2-1-58tvq468j3.pdf

Traces, Cross-Ratios and 2-Generator Subgroups of SU(2,1)

We embed the Teichmuller space of the once punctured torus T(1,1) into the set of conjugacy classes of groups generated by three anti-holomorphic involutions I1, I2 and I3 (Lagrangian triangle groups), acting on the complex hyperbolic plane HC. We deform this embedding, and obtain a three dimensional family E of discrete, faithful and type preserving representations of the fundamental group of the once punctured torus. AMS classification 51M10, 32M15, 22E40

/pdf/the-punctured-torus-and-lagrangian-triangle-groups-in-pu-2-1-1vxv1h7pzg.pdf

The punctured torus and Lagrangian triangle groups in PU(2,1)

We study subgroups of ${\rm PU}(2,1)$ generated by two non-commuting 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 $\mathbb{R}^2$ . By considering the action on complex hyperbolic space $\mathbf{H}^2_{\mathbb{C}}$ 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 $(3,3,\infty)$-triangle groups. We also consider a particular group on the boundary of the disc ${\mathcal Z}$ where the commutator $[A,B]$ 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.

/pdf/a-complex-hyperbolic-riley-slice-491g9f670m.pdf

A complex hyperbolic Riley slice

We provide a concrete criterion to determine whether or not two given elements of PU(2,1) can be written as products of real reflections, with one reflection in common. As an application, we show that the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ with $d=1,2,3,7,11$ are generated by real reflections up to index 1, 2, 4 or 8.

https://hal.archives-ouvertes.fr/hal-00916997/document

Real reflections, commutators and cross-ratios in complex hyperbolic space

We study subgroups of PU(2,1) generated by two non-commuting unipotent maps A and B whose product AB is also unipotent. We call U the set of conjugacy classes of such groups. We provide a set of coordinates on U that make it homeomorphic to R2 . By considering the action on complex hyperbolic space H2C of groups in U, we describe a two dimensional disc Z in U that parametrises a family of discrete groups. As a corollary, we give a proof of a conjecture of Schwartz for (3,3,∞)-triangle groups. We also consider a particular group on the boundary of the disc Z where the commutator [A,B] 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.

/pdf/a-complex-hyperbolic-riley-slice-3qatbhpget.pdf

Pierre Will

Papers

Traces, Cross-Ratios and 2-Generator Subgroups of SU(2,1)

The punctured torus and Lagrangian triangle groups in PU(2,1)

A complex hyperbolic Riley slice

Real reflections, commutators and cross-ratios in complex hyperbolic space

A complex hyperbolic Riley slice