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Virginie Charette

Publications and source records attributed to Virginie Charette.

16 recordsLinked to original sources
arXiv2019

Einstein tori and crooked surfaces

In hyperbolic space, the angle of intersection and distance classify pairs of totally geodesic hyperplanes. A similar algebraic invariant classifies pairs of hyperplanes in the Einstein universe. In dimension 3, symplectic splittings of a 4-dimensional real symplectic vector space model Einstein hyperplanes and the invariant is a determinant. The classification contributes to a complete disjointness criterion for crooked surfaces in the 3-dimensional Einstein universe.

math.DG↗
arXiv2016

McShane-type Identities for Affine Deformations

We derive an identity for Margulis invariants of affine deformations of a complete orientable one-ended hyperbolic sur- face following the identities of McShane, Mirzakhani and Tan- Wong-Zhang. As a corollary, a deformation of the surface which infinitesimally lengthens all interior simple closed curves must in- finitesimally lengthen the boundary.

math.GT↗
arXiv2016

Bisectors determining unique pairs of points in the bidisk

Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries (spherical, Euclidean, hyperbolic, complex hyperbolic, to name a few) bisectors do not uniquely determine a pair of points, in the following sense\,: completely different sets of points share a common bisector. The above examples of this non-uniqueness are all rank $1$ symmetric spaces. However, as we show in this paper, bisectors in the usual $L^2$ metric are such for a unique pair of points in the rank $2$ geometry $\mathbb{H}^2 \times\mathbb{H}^2$.

math.DG↗
arXiv2015

Proper affine deformation spaces of two-generator Fuchsian groups

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 decomposes into polyhedra bounded by crooked planes, corresponding to an ideal triangulation of S. This paper classifies and analyzes the structure of crooked ideal triangles, which play the same role for Margulis spacetimes as ideal triangles play for hyperbolic surfaces. This extends our previous work on affine deformations of three-holed sphere and two-holed cross surfaces.

math.DG↗
arXiv2014

Fundamental domains in the Einstein Universe

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 in the 3-dimensional Einstein Universe. As an application, we can construct explicit examples of groups acting properly on an open subset of that space.

math.DG↗
arXiv2013

Foliations of Minkowski $2+1$ spacetime by crooked planes

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.

math.DG↗
arXiv2013

Crooked Halfspaces

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 parallelism classes of timelike lines, or particles, in a crooked halfspace is a geodesic halfplane in the hyperbolic plane. Every point in an open crooked halfspace lies on a particle. The correspondence between crooked halfspaces and halfplanes in hyperbolic 2-space preserves the partial order defined by inclusion, and the involution defined by complementarity. We find conditions for when a particle lies completely in a crooked half space. We revisit the disjointness criterion for crooked planes developed by Drumm and Goldman in terms of the semigroup of translations preserving a crooked halfspace. These ideas are then applied to describe foliations of Minkowski space by crooked planes.

math.DG↗
arXiv2012

Finite-sided deformation spaces of complete affine 3-manifolds

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 to a two-holed cross-surface, that is, the complement of two disjoint discs in the real projective plane. We show that every such manifold is homeomorphic to a solid handlebody of genus two, and admits a fundamental polyhedron bounded by crooked planes. Furthermore, the deformation space is a bundle of convex quadrilateral cones over the space of marked hyperbolic structures. The sides of each quadrilateral cone are defined by invariants of the two boundary components and the two orientation-reversing simple curves. The two-holed cross-surface, together with the three-holed sphere, are the only topologies for which the deformation space of complete affine structures is finite-sided.

math.GT↗
arXiv2010

Affine Schottky Groups and Crooked Tilings

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 1980's, answering a question raised by Milnor in 1977. This paper expounds Drumm's result, at least in the case of Fuchsian Schottky groups (that is, when the group contains no parabolic elements).

math.DG↗
arXiv2009

Stretching three-holed spheres and the Margulis invariant

This paper applies the authors' forthcoming work, "Affine deformations of a three-holed sphere" in Lorentzian geometry to prove a result in hyperbolic geometry. Namely, an infinitesimal deformation of a hyperbolic structure of a three-holed sphere which infinitesimally lengthens the three boundary components infinitesimally lengthens every closed geodesic. The proof interprets the derivative of the geodesic length function as the Margulis invariant (signed marked Lorentzian length spectrum) of the corresponding affine deformation. The aforementioned results imply that the affine deformation is proper, and hence by Margulis's Opposite Sign Lemma, every closed geodesic infinitesimmaly lengthens.

math.DG↗
arXiv2009

Affine deformations of a three-holed sphere

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 two opposite octants in R^3. Furthermore every M admits a fundamental polyhedron bounded by crooked planes. Therefore M is homeomorphic to an open solid handlebody of genus two. As an explicit application of this theory, we construct proper affine deformations of an arithmetic Fuchsian group inside Sp(4,Z).

math.DG↗
arXiv2007

A primer on the (2+1) Einstein universe

The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension $2 + 1$, in which there is a rich interplay with symplectic geometry.

math.DG↗
arXiv2003

Non-proper Actions of the Fundamental Group of a Punctured Torus

Given an affine isometry of $\R^3$ with hyperbolic linear part, its Margulis invariant measures signed Lorentzian displacement along an invariant spacelike line. In order for a group generated by hyperbolic isometries to act properly on $\R^3$, the sign of the Margulis invariant must be constant over the group. We show that, in the case when the linear part is the fundamental group of a punctured torus, positivity of the Margulis invariant over any finite generating set does not imply that the group acts properly. This contrasts with the case of a pair of pants, where it suffices to check the sign of the Margulis invariant for a certain triple of generators.

math.DG↗
arXiv2003

Strong marked isospectrality of affine Lorentzian groups

The Margulis invariant is a function defined on a group of Lorentzian transformations $G$ acting on Minkowski space $\R^{2,1}$, that contains no elliptic elements. The spectrum of $G$ is the sequence of values of the Margulis invariant for all its elements. If the underlying linear group of $G$ is fixed, Drumm and Goldman proved that the spectrum defines the translational part completely. In this note, we strengthen this result by showing that isospectrality holds for any free product of cyclic groups of given rank, up to conjugation in the group of affine transformations of $R^{2,1}$, as long as it is non-radiant and that its linear part is discrete and non-elementary. In particular, isospectrality holds when the linear part is a Schottky group.

math.DG↗
arXiv2002

Closed Timelike Curves in Flat Lorentz Spacetimes

We consider the region of closed timelike curves (CTC's) in three-dimensional flat Lorentz spacetimes. The interest in this global geometrical feature goes beyond the purely mathematical. Such spacetimes may be considered lower-dimensional toy models of sourceless Einstein gravity or cosmology. In particular, our interest in this note will be to find the set free of CTC's for $E/<γ>$, where $E$ is modeled on Minkowski space and $γ$ is a Poincaré transformation. We describe the set free of CTC's where $γ$ is hyperbolic, parabolic and elliptic.

math.DG↗