33 research outputs found
Non-arithmetic lattices and the Klein quartic
We give an algebro-geometric construction of some of the non-arithmetic ball
quotients constructed by the author, Parker and Paupert. The new construction
reveals a relationship between the corresponding orbifold fundamental groups
and the automorphism group of the Klein quartic, and also with groups
constructed by Barthel-Hirzebruch-H\"ofer and Couwenberg-Heckman-Looijenga
A new non-arithmetic lattice in PU(3,1)
We study the arithmeticity of the Couwenberg-Heckman-Looijenga lattices in
PU(n,1), and show that they contain a non-arithmetic lattice in PU(3,1) which
is not commensurable to the non-arithmetic Deligne-Mostow lattice in PU(3,1)
A 1-parameter family of spherical CR uniformizations of the figure eight knot complement
We describe a simple fundamental domain for the holonomy group of the
boundary unipotent spherical CR uniformization of the figure eight knot
complement, and deduce that small deformations of that holonomy group (such
that the boundary holonomy remains parabolic) also give a uniformization of the
figure eight knot complement. Finally, we construct an explicit 1-parameter
family of deformations of the boundary unipotent holonomy group such that the
boundary holonomy is twist-parabolic. For small values of the twist of these
parabolic elements, this produces a 1-parameter family of pairwise
non-conjugate spherical CR uniformizations of the figure eight knot complement
On the geometry of a Picard modular group
We study geometric properties of the action of the Picard modular group
on the complex hyperbolic plane
, where denotes the ring of algebraic integers
in . We list conjugacy classes of maximal finite
subgroups in and give an explicit description of the Fuchsian
subgroups that occur as stabilizers of mirrors of complex reflections in
. As an application, we describe an explicit torsion-free subgroup of
index in
Almost quarter-pinched K\"ahler metrics and Chern numbers
We prove that compact K\"ahler manifolds whose sectional curvatures are close
to 1/4-pinched have ratios of Chern numbers close to the corresponding ratios
of a complex hyperbolic space form. We deduce that the Mostow-Siu surfaces (and
their three-dimensional analogues constructed by the first author) do not admit
K\"ahler metrics with pinching close to 1/4
On subgroups of finite index in complex hyperbolic lattice triangle groups
We study several explicit finite index subgroups in the known complex
hyperbolic lattice triangle groups, and show some of them are neat, some of
them have positive first Betti number, some of them have a homomorphisms onto a
non-Abelian free group. For some lattice triangle groups, we determine the
minimal index of a neat subgroup. Finally, we answer a question raised by
Stover and describe an infinite tower of neat ball quotients all with a single
cusp
Volumes of 3-ball quotients as intersection numbers
We give an explicit description of the 3-ball quotients constructed by
Couwenberg-Heckman-Looijenga, and deduce the value of their orbifold Euler
characteristics. For each lattice, we also give a presentation in terms of
generators and relations.Comment: arXiv admin note: substantial text overlap with arXiv:1710.0446
On spherical CR uniformization of 3-manifolds
International audienceWe consider the three discrete representations in the Falbel-Koseleff-Rouillier census where the peripheral subgroups have cyclic 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 representation with cyclic unipotent boundary holonomy is not in general the holonomy of a spherical CR uniformization of
Forgetful maps between Deligne-Mostow ball quotients
We study forgetful maps between Deligne-Mostow moduli spaces of weighted
points on P^1, and classify the forgetful maps that extend to a map of
orbifolds between the stable completions. The cases where this happens include
the Livn\'e fibrations and the Mostow/Toledo maps between complex hyperbolic
surfaces. They also include a retraction of a 3-dimensional ball quotient onto
one of its 1-dimensional totally geodesic complex submanifolds
On the universal cover of certain exotic KĂ€hler surfaces of negative curvature
International audienc