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Journals and Conferences
We prove that the last discrete deformation of the RFuchsian (4,4,4)-triangle group in PU(2, 1) is a cocompact arithmetic lattice. We also describe an experimental method for finding the combinatorics of a Dirichlet fundamental domain, and apply it to the lattice in question.
We show that the figure eight knot complement admits a uniformizable spherical CR structure, i.e. it occurs as the manifold at infinity of a complex hyperbolic orbifold. The uniformization is unique provided we require the peripheral subgroups to have unipotent holonomy.
We describe an experimental method for studying the combinatorics of Dirichlet domains in the complex hyperbolic plane, based on numerical and graphical techniques. We apply our techniques to the complex reflection groups that appear in Mostow’s seminal paper on the subject, and list a number of corrections to the combinatorics of the Dirichlet domains.
We construct examples of three-dimensional compact Kähler manifolds with negative curvature, not covered by the ball. Our manifolds are obtained as a natural generalization of the two-dimensional examples discovered by Mostow and Siu, using their description in terms of monodromy covers of hypergeometric functions. Each example is obtained from a… (More)
We give a new construction of fundamental domains in H C for the action of certain lattices in PU(2, 1) defined by Mostow. The polyhedra are given a natural geometric description starting from certain fixed points of elliptic elements. Among the advantages over Dirichlet domains, we gain a simplification of the combinatorics and obtain proofs using mainly… (More)
We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livné fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also include a retraction of a 3-dimensional… (More)
The goal of this paper is to give a conjectural census of complex hyperbolic sporadic triangle groups. We prove that only finitely many of these sporadic groups are lattices. We also give a conjectural list of all lattices among sporadic groups, and for each group in the list we give a conjectural group presentation, as well as a list of cusps and… (More)
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)
We consider the discrete representations of 3-manifold groups into PU(2, 1) that appear in the Falbel-Koseleff-Rouillier, such that the peripheral subgroups have cyclic unipotent holonomy. We show that two of these representations have conjugate images, even though they represent different 3-manifold groups. This illustrates the fact that a discrete… (More)
Given n ∈ Z and ε > 0, we prove that there exists δ = δ(ε, n) > 0 such that the following holds: If (M, g) is a compact Kähler n-manifold whose sectional curvatures K satisfy