55,706 research outputs found
Complex hyperbolic triangle groups
The theory of complex hyperbolic discrete groups is still in its childhood
but promises to grow into a rich subfield of geometry. In this paper I will
discuss some recent progress that has been made on complex hyperbolic
deformations of the modular group and, more generally, triangle groups. These
are some of the simplest nontrivial complex hyperbolic discrete groups. In
particular, I will talk about my recent discovery of a closed real hyperbolic
3-manifold which appears as the manifold at infinity for a complex hyperbolic
discrete group
Geometry and topology of complex hyperbolic and CR-manifolds
We study geometry, topology and deformation spaces of noncompact complex
hyperbolic manifolds (geometrically finite, with variable negative curvature),
whose properties make them surprisingly different from real hyperbolic
manifolds with constant negative curvature. This study uses an interaction
between K\"ahler geometry of the complex hyperbolic space and the contact
structure at its infinity (the one-point compactification of the Heisenberg
group), in particular an established structural theorem for discrete group
actions on nilpotent Lie groups
Discrete conformal maps and ideal hyperbolic polyhedra
We establish a connection between two previously unrelated topics: a
particular discrete version of conformal geometry for triangulated surfaces,
and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated
surfaces are considered discretely conformally equivalent if the edge lengths
are related by scale factors associated with the vertices. This simple
definition leads to a surprisingly rich theory featuring M\"obius invariance,
the definition of discrete conformal maps as circumcircle preserving piecewise
projective maps, and two variational principles. We show how literally the same
theory can be reinterpreted to addresses the problem of constructing an ideal
hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables
us to derive a companion theory of discrete conformal maps for hyperbolic
triangulations. It also shows how the definitions of discrete conformality
considered here are closely related to the established definition of discrete
conformality in terms of circle packings.Comment: 62 pages, 22 figures. v2: typos corrected, references added and
updated, minor changes in exposition. v3, final version: typos corrected,
improved exposition, some material moved to appendice
Partialy Paradoxist Smarandache Geometries
A paradoxist Smarandache geometry combines Euclidean, hyperbolic, and elliptic geometry into one space along with other non-Euclidean behaviors oflines that would seem to require a discrete space. A class of continuous spaces is presented here together with specific examples that emibit almost all of these phenomena and suggest the prospect of a continuous paradoxist geometry
Representations of polygons of finite groups
We construct discrete and faithful representations into the isometry group of
a hyperbolic space of the fundamental groups of acute negatively curved
even-sided polygons of finite groups.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper43.abs.htm
On Weingarten transformations of hyperbolic nets
Weingarten transformations which, by definition, preserve the asymptotic
lines on smooth surfaces have been studied extensively in classical
differential geometry and also play an important role in connection with the
modern geometric theory of integrable systems. Their natural discrete analogues
have been investigated in great detail in the area of (integrable) discrete
differential geometry and can be traced back at least to the early 1950s. Here,
we propose a canonical analogue of (discrete) Weingarten transformations for
hyperbolic nets, that is, C^1-surfaces which constitute hybrids of smooth and
discrete surfaces "parametrized" in terms of asymptotic coordinates. We prove
the existence of Weingarten pairs and analyse their geometric and algebraic
properties.Comment: 41 pages, 30 figure
Some open questions on anti-de Sitter geometry
We present a list of open questions on various aspects of AdS geometry, that
is, the geometry of Lorentz spaces of constant curvature -1. When possible we
point out relations with homogeneous spaces and discrete subgroups of Lie
groups, to Teichm\"uller theory, as well as analogs in hyperbolic geometry.Comment: Not a research article in the usual sense but rather a list of open
questions. 19 page
Characteristic Algebras of Fully Discrete Hyperbolic Type Equations
The notion of the characteristic Lie algebra of the discrete hyperbolic type
equation is introduced. An effective algorithm to compute the algebra for the
equation given is suggested. Examples and further applications are discussed.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
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