Crooked planes are piecewise linear surfaces that were introduced by Drumm in the early 1990s to construct fundamental domains for properly discontinuous actions of free groups on Minkowski 3-space. In a previous paper, we introduced analogues of these surfaces, called AdS crooked planes, in the 3-dimensional anti-de S…
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48 results for “Crooked fundamental domains”
We will discuss fundamental domains for actions of discrete groups on the 3-dimensional Einstein Universe. These will be bounded by crooked surfaces, which are conformal compactifications of surfaces that arise in the construction of Margulis spacetimes. We will show that there exist pairwise disjoint crooked surfaces …
The study examines extensions of a projective plane minus two discs in flat affine Lorentzian 3-manifolds.
problem Classifying proper actions of affine Coxeter extensions on Minkowski space.
method Investigates double extensions of fundamental groups and their actions on Minkowski space.
result Identifies proper actions of the double extension that do not admit crooked fundamental domains.
The paper defines cocycles for positive Anosov representations and constructs affine actions with bounded fundamental domains.
problem Positive Anosov representations into .
method Definition of cocycles and construction of affine actions with fundamental domains.
result Quotient manifolds are homeomorphic to handlebodies.
Crooked planes were defined by Drumm to bound fundamental polyhedra in Minkowski space for Margulis spacetimes. They were extended by Frances to closed polyhedral surfaces in the conformal compactification of Minkowski space (Einstein space) which we call crooked surfaces. The conformal model of anti-de Sitter space is…
We study strip deformations of convex cocompact hyperbolic surfaces, defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface. We prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation, in…
Crooked planes in 3D Minkowski space can be foliated.
problem Understanding foliations between crooked planes in 3D Minkowski space.
method Showed that any two disjoint crooked planes are leaves of a crooked foliation.
result Answered a question about foliations between crooked planes in 3D Minkowski space.
Crooked Halfspacesmath.DG
We develop the Lorentzian geometry of a crooked halfspace in 2+1-dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving an explicit slice for the action of the automorphism group. The set of parall…
A Margulis spacetime is a complete flat Lorentzian 3-manifold M with free fundamental group. Associated to M is a noncompact complete hyperbolic surface S homotopy-equivalent to M. The purpose of this paper is to classify Margulis spacetimes when S is homeomorphic to a one-holed torus. We show that every such M decompo…
Given a regular curve in Minkowski spacetime, we describe necessary and sufficient conditions for this curve to admit a family of pairwise-disjoint crooked planes. Using this criterion, we describe crooked foliations along orbit curves of one-parameter groups of Lorentzian isometries.
Classifies pairs of hyperplanes in Einstein universe, contributing to crooked surfaces classification.
problem Classifying pairs of hyperplanes in the Einstein universe.
method Algebraic invariant and symplectic splittings model Einstein hyperplanes.
result Contributes to a complete disjointness criterion for crooked surfaces.
FEAT estimates free energy using adaptive transports.
problem Estimating free energy across scientific domains.
method Uses learned transports and stochastic interpolants.
result Provides consistent, minimum-variance estimators.
New groups discovered with unique properties in a specific space.
problem Finding new discrete subgroups with special properties in a mathematical space.
method Proved by showing groups play ping-pong on cones, related to crooked surfaces.
result Infinite family of discrete subgroups with remarkable properties in .
Associated to every complete affine 3-manifold M with nonsolvable fundamental group is a noncompact hyperbolic surface S. We classify such complete affine structures when Sigma is homeomorphic to a three-holed sphere. In particular, for every such complete hyperbolic surface Sigma, the deformation space identifies with…
In his 1990 doctoral thesis, Todd Drumm showed that proper affine deformations of free Fuchsian groups could be constructed as Schottky groups using a new family of hypersurfaces called "crooked planes." The existence of proper affine deformations of Fuchsian Schottky groups was demonstrated by Margulis in the early 19…
A survey on the recent work of Danciger, Guéritaud and Kassel on Margulis space-times and complete anti-de Sitter space-times. Margulis space-times are quotients of the 3-dimensional Minkowski space by (non-abelian) free groups acting propertly discontinuously. Goldman, Labourie and Margulis have shown that they are de…
A Margulis spacetime is a complete affine 3-manifold M with nonsolvable fundamental group. Associated to every Margulis spacetime is a noncompact complete hyperbolic surface S. We show that every Margulis spacetime is orientable, even though S may be nonorientable. We classify Margulis spacetimes when S is homeomorphic…
The paper studies fundamental domains in H^3 and their associated polyhedra.
problem Understanding the relationship between polyhedra and groups associated with fundamental domains in H^3.
method Analyzes torsion-free groups and edge classes of abstract polyhedra, proving results about group properties and edge classes.
result Classifies fundamental domains on the cube with torsion-free groups and provides insights into polyhedra and groups.
The paper defines fundamental domains for Fuchsian groups in hyperbolic space.
problem Identifying fundamental domains for Fuchsian groups.
method Provides a condition for sets to be fundamental domains via hyperbolic space.
result A necessary and sufficient condition for fundamental domains in .
Fundamental gap vanishes for convex domains in hyperbolic space.
problem Behavior of fundamental gap in convex domains in hyperbolic space.
method Proof for Laplace operator with Dirichlet boundary conditions.
result Fundamental gap can be arbitrarily small for domains of any diameter.
Research explores hyperbolic space groups and their fundamental domains.
problem Investigating fundamental domains of space groups in hyperbolic spaces.
method Analyzing symmetries of fundamental polyhedra and considering edge conditions.
result Identifies edge conditions for simplicial fundamental domains of Family F12.
Chirality for crooked curvescond-mat.soft
Tensor measures chirality for curves, even those with rough edges.
problem Quantifying chirality for complex, possibly irregular curves.
method Developed a tensorial chirality measure for rigid filaments and curves.
result A curve's chirality can be determined by its twist about perpendicular axes.
Study shows fundamental gap of horoconvex domains in hyperbolic space has no positive lower bound.
problem Understanding the fundamental gap of horoconvex domains in hyperbolic space.
method Analysis of fundamental gap of geodesic balls as radius goes to infinity.
result Product of fundamental gap and square of diameter has no positive lower bound for horoconvex domains.
Constant potentials do not minimise the fundamental gap on convex domains in hyperbolic spacemath.AP
New convex domains in hyperbolic space can have lower fundamental gap than constant potentials.
problem Finding convex domains with lower fundamental gap than constant potentials.
method Constructing specific convex domains and potentials with controlled eigenfunctions.
result Fundamental gap of can be strictly smaller than for convex domains.
Sharp fundamental gap estimate proved for convex domains on spheres.
problem Proving a sharp fundamental gap estimate for convex domains on spheres.
method Proving super log-concavity of the first eigenfunction.
result Sharp fundamental gap bound of for .
Proves a fundamental gap lower bound for horoconvex domains in hyperbolic space.
problem Proving a fundamental gap lower bound for horoconvex domains in hyperbolic space.
method Reduces the problem to a radial-height problem, compares Dirichlet forms with angular operators, and uses Green estimates.
result Establishes a polynomial \(D^{-3}\) scale fundamental gap lower bound.
Researchers create a fundamental domain for all Deligne-Mostow lattices in PU(2,1).
problem Building a fundamental domain for all Deligne-Mostow lattices in PU(2,1).
method Using Thurston's approach, the team constructed a fundamental domain for lattices with 2-fold symmetry, completing the list of commensurability classes.
result Fundamental domains were created for all commensurability classes of Deligne-Mostow lattices in PU(2,1).
Researchers find gap between eigenvalues for convex domains in hyperbolic plane.
problem Estimating the gap between the first and second Dirichlet eigenvalues for convex domains in hyperbolic space.
method Computed fundamental gap for a family of convex domains in .
result For some convex domains, the gap , where is the diameter.
Negative curvature restricts the gap between the first and second eigenvalues of convex domains.
problem The fundamental gap of convex domains is limited by negative curvature.
method Adapted from Bourni et. al. (2022) for Riemannian manifolds with negative sectional curvature.
result The product of the fundamental gap and the square of the diameter can be arbitrarily small in domains with negative curvature.
New fundamental domain for Hilbert-Blumenthal cusp shapes.
problem Understanding the geometry of Hilbert-Blumenthal surfaces at cusps.
method Constructing a fundamental domain using Dirichlet domains and deformations of lattices.
result Explicitly describes the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism.
We show the existence of isometric (or Ford) fundamental regions for a large class of subgroups of the isometry group of any rank one Riemannian symmetric space of noncompact type. The proof does not use the classification of symmetric spaces. All hitherto known existence results of isometric fundamental regions and do…
The main result of this paper is a construction of fundamental domains for certain group actions on Lorentz manifolds of constant curvature. We consider the simply connected Lie group G~, the universal cover of the group SU(1,1) of orientation-preserving isometries of the hyperbolic plane. The Killing form on the Lie g…
The study shows that the second fundamental form is intrinsic under certain conditions in space forms.
problem Understanding the intrinsic nature of the second fundamental form in space forms.
method Proving the intrinsic nature of the normalized second fundamental form under specific conditions.
result The normalized second fundamental form is intrinsic if for some .
The study improves fundamental gap estimates for surfaces with non-constant positive curvature.
problem Estimating the fundamental gap for surfaces with non-constant positive curvature.
method Using a two-point maximum principle, the study establishes log-concavity and fundamental gap estimates.
result Corresponding log-concavity and fundamental gap estimates for surfaces with non-constant positive curvature are derived.
The article proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
problem Proving a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
method Establishing conformal log-concavity estimates for the first eigenfunction.
result Proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.
Method constructs fundamental domains for Picard modular groups.
problem Classify and understand torsion elements in Picard modular groups.
method Systematic construction of coarse fundamental domains.
result Classification of conjugacy classes of torsion elements.
Sharp estimates for heat flow on nonconvex domains.
problem Quantitative estimates for heat flow on nonconvex domains.
method Sharp gradient and transport estimates with novel dependence on time.
result Equivalent characterization of lower bound on second fundamental form.
Algorithm finds Dirichlet domains for hyperbolic surfaces.
problem Computing explicit Dirichlet domains for hyperbolic surfaces.
method Algorithm based on side pairings of fundamental polygons, using geometric-topological data structures.
result Algorithm runs in polynomial time, dependent on surface perimeter and genus.
CMCD sampler connects transport and variational inference for efficient sampling.
problem Efficient sampling and generative modeling in Bayesian computation.
method Developed a principled framework using divergences on path space, CMCD sampler with adaptive dynamics.
result CMCD sampler outperforms competing approaches across various experiments.
Study -Fuchsian subgroups of non-arithmetic lattices.
problem Understand structure and fundamental domains of -Fuchsian subgroups.
method General procedure to analyze structure and show fundamental domains lie on a complex geodesic.
result Fundamental domains of -Fuchsian subgroups lie on a complex geodesic homeomorphic to the unit disk.
Construct Schottky subgroups for maximal representations in symmetric spaces.
problem Maximal representations of surface groups into Sp(2n, R).
method Construction of Schottky type subgroups of automorphism groups.
result Schottky subgroups correspond to maximal representations of surface groups.
Lower bounds for quaternionic hyperbolic orbifold volumes found.
problem Finding explicit lower bounds for quaternionic hyperbolic orbifold volumes.
method Using H. C. Wang's radius bound for fundamental domains of semisimple Lie groups.
result Explicit lower bound for quaternionic hyperbolic orbifold volumes depending only on dimension.
At most 27 length inequalities define Maskit's fundamental domain for the modular group in genus 2math.GT
In recently published work Maskit constructs a fundamental domain D_g for the Teichmueller modular group of a closed surface S of genus g>1. Maskit's technique is to demand that a certain set of 2g non-dividing geodesics C_{2g} on S satisfies certain shortness criteria. This gives an a priori infinite set of length ine…
Log-concavity of eigenfunctions on curved surfaces is proven, leading to fundamental gap estimates.
problem Proving log-concavity of eigenfunctions on curved surfaces.
method Analyzing the Laplacian eigenfunctions on positively curved surfaces.
result Strong log-concavity of the first eigenfunction on positively curved surfaces.
Small deformations of a specific type of Lorentzian space-time preserve its structure.
problem Deformations of Margulis space-times with parabolics.
method Use of previous work on compactification and Carrière's decomposition of the space-time.
result Sufficiently small deformations of the group Γ still act properly on the space-time.
Paper constructs two series of Lorentz bi-quotients from polyhedra.
problem Creating fundamental domains for Lorentz bi-quotients.
method Explicit construction of polyhedral fundamental domains.
result Two infinite series of Lorentz bi-quotients constructed.
We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas.