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1 Complex hyperbolic space There are several ways to generalise the hyperbolic plane and its isometry group to objects in higher dimensions. Perhaps the most familiar is (real) hyperbolic three space, popularised by the work of Thurston [14]. The Poincaré disc and half plane models of the hyperbolic plane naturally come with a complex structure and it is… (More)

We consider symmetric complex hyperbolic triangle groups generated by three complex reeec-tions with angle 2=p. We restrict our attention to those groups where certain words are elliptic. Our goal is to nd necessary conditions for such a group to be discrete. The main application we have in mind is that such groups are candidates for non-arithmetic lattices… (More)

- J. R. Parker
- 2005

Let π 1 be the fundamental group of a closed surface Σ of genus g > 1. One of the fundamental problems in complex hyperbolic geometry is to find all discrete, faithful, geometrically finite and purely loxodromic representations of π 1 into SU(2, 1), (the triple cover of) the group of holomorphic isometries of H 2 C. In particular , given a discrete,… (More)

- Elisha Falbel, Gábor Francsics, John R. Parker
- 2009

We give a construction of a fundamental domain for the group PU(2, 1, Z[i]). That is the group of holomorphic isometries of complex hyperbolic space with coefficients in the Gaussian ring of integers Z[i]. We obtain from that construction a presentation of that lattice and relate it, in particular, to lattices constructed by Mostow.

We produce a family of new, non-arithmetic lattices in PU(2, 1). All previously known examples were commensurable with lattices constructed by Picard, Mostow, and Deligne– Mostow, and fell into 9 commensurability classes. Our groups produce 5 new distinct commensurability classes. Most of the techniques are completely general, and provide efficient… (More)

The purpose of this paper is twofold. First, we give a survey of the known methods of constructing lattices in complex hyperbolic space. Secondly, we discuss some of the lattices constructed by Deligne and Mostow and by Thurston in detail. In particular, we give a unified treatment of the constructions of fundamental domains and we relate this to other… (More)

A complex hyperbolic quasi-Fuchsian group is a discrete, faithful, type preserving and geometrically finite representation of a surface group as a subgroup of the group of holomorphic isometries of complex hyperbolic space. Such groups are direct complex hyperbolic generali-sations of quasi-Fuchsian groups in three dimensional (real) hyperbolic geometry. In… (More)

- John R. Parker
- 2007

A complex hyperbolic triangle group is the group of complex hyperbolic isometries generated by complex involutions fixing three complex lines in complex hyperbolic space. Such a group is called equilateral if there is an isometry of order three that cyclically permutes the three complex lines. We consider equilateral triangle groups for which the product of… (More)

Let Σ be a closed, orientable surface of genus g. It is known that the SU(2, 1) representation variety of π1(Σ) has 2g − 3 components of (real) dimension 16g − 16 and two components of dimension 8g−6. Of special interest are the totally loxodromic, faithful (that is quasi-Fuchsian) representations. In this paper we give global real analytic coordinates on a… (More)

- SHIGEYASU KAMIYA, JOHN R. PARKER, Shigeyasu Kamiya, John R. Parker
- 2004

We give a version of Shimizu's lemma for groups of complex hyperbolic isometries one of whose generators is a parabolic screw motion. Suppose that G is a discrete group containing a parabolic screw motion A and let B be any element of G not fixing the fixed point of A. Our result gives a bound on the radius of the isometric spheres of B and B −1 in terms of… (More)