Stretching three-holed spheres and the Margulis invariant (2009)
Charette, Virginie, Drumm, Todd A., Goldman, William M.
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...
Affine deformations of a three-holed sphere (2009)
Charette, Virginie, Drumm, Todd A., Goldman, William M.
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...
Higgs Bundles and Geometric Structures on Surfaces (2008)
This paper concerns the relationship between locally homogeneous geometric structures on topological surfaces and the moduli of polystable Higgs bundles on Riemann surfaces, due to Hitchin and...
The Geometry of Crooked Planes (2007)
Todd A. Drumm, William M. Goldman
. Crooked planes are polyhedra used to construct fundamental polyhedra for discrete groups of Lorentz isometries acting properly on Minkowski (2+1)-space. This paper explores intersections of crooked...
RECURRENT GEODESICS IN FLAT LORENTZ 3-MANIFOLDS (2007)
Virginie Charette, William M. Goldman, Catherine A
Abstract. We introduce the notion of recurrent geodesic rays in a complete at Lorentz 3-manifold. We completely classify the dynamical behavior of geodesics in cyclic quotients, and apply this classi...
William M. Goldman, William M. Goldman
Let G = SU(n) and π the fundamental group of a closed oriented surface S. Let Hom(π, G) be the moduli space of conjugacy classes of representations π − → G and let X ⊂ Hom(π, G) be the...
The Margulis Invariant of Isometric Actions on Minkowski (2+1)-Space (2007)
Abstract. Let E denote an affine space modelled on Minkowski (2+1)-space E and let Γ be a group of isometries whose linear part L(Γ) is a purely hyperbolic subgroup of SO 0 (2, 1). Margulis has...
A primer on the (2+1) Einstein universe (2007)
Barbot, Thierry, Charette, Virginie, Drumm, Todd, Goldman, William M., Melnick, Karin
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...
Notes on a paper of Mess (2007)
Andersson, Lars, Barbot, Thierry, Benedetti, Riccardo, Bonsante, Francesco, Goldman, William M., Labourie, François, ...
These notes are a companion to the article "Lorentz spacetimes of constant curvature" by Geoffrey Mess, which was first written in 1990 but never published. Mess' paper will appear together with...
Notes on a paper of Mess (2007)
Andersson, Lars, Barbot, Thierry, Benedetti, Riccardo, Bonsante, Francesco, Goldman, William M., Labourie, François, ...
These notes are a companion to the article "Lorentz spacetimes of constant curvature" by Geoffrey Mess, which was first written in 1990 but never published. Mess' paper will appear together with...
Mapping Class Group Dynamics on Surface Group Representations (2005)
Deformation spaces Hom($\pi$,G)/G of representations of the fundamental group $\pi$ of a surface $\Sigma$ in a Lie group $G$ admit natural actions of the mapping class group $Mod_\Sigma$, preserving...
The Mapping Class Group acts reducibly on SU(n)-character varieties (2005)
When $G$ is a connected compact Lie group, and $\pi$ is a closed surface group, then $Hom(\pi,G)$ contains an open dense $Out(\pi)$-invariant subset which is a smooth symplectic manifold. This...
An ergodic action of the outer automorphism group of a free group (2005)
For n>2, the action of the outer automorphism group of the rank n free group F_n on the SU(2)-character variety Hom(F_n,SU(2))/SU(2)$ is ergodic with respect to the Lebesgue measure class.
Energy of Twisted Harmonic Maps of Riemann Surfaces (2005)
Goldman, William M., Wentworth, Richard A.
The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface $S$ is a function $E_\rho$ on Teichm\"uller space $\Teich$ which is a qualitative...
Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces (2004)
Goldman, William M., Xia, Eugene Z.
This expository paper details the theory of rank one Higgs bundles over a closed Riemann surface X and their relationship to representations of the fundamental group of X. We construct an equivalence...
An exposition of results of Fricke (2004)
This paper gives an elementary proof of the result that the conjugacy classes of pairs (X,Y) of unimodular 2x2 complex matrices is an affine 3-space, parametrized by the traces of X, Y and XY....
Homological action of the modular group on some cubic moduli spaces (2004)
Goldman, William M., Neumann, Walter D.
We describe the action of the automorphism group of the complex cubic x^2+y^2+z^2-xyz-2 on the homology of its fibers. This action includes the action of the mapping class group of a punctured torus...
Is the deformation space of complete affine structures on the 2-torus smooth? (2004)
Baues, Oliver, Goldman, William M.
Periods of parallel exterior forms define natural coordinates on the deformation space of complete affine structures on the two-torus. These coordinates define a differentiable structure on this...
The Modular Group Action on Real SL(2)-characters of a One-Holed Torus (2003)
The group Gamma of automorphisms of the polynomial kappa(x,y,z) = x^2 + y^2 + z^2 - xyz -2 is isomorphic to PGL(2,Z) semi-direct product with (Z/2+Z/2). For t in R, Gamma-action on ktR =...
The complex-symplectic geometry of SL(2,C)-characters over surfaces (2003)
The SL(2)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group G of M. Using the ergodicity of G on the SU(2)-character...
The Complex-Symplectic Geometry of SL(2, C)-Characters over Surfaces (2003)
The SL(2, C)-character variety X of a closed surface M enjoys a natural complex-symplectic structure invariant under the mapping class group # of M . Using the ergodicity of # on the SU(2)-character...
Isospectrality of Flat Lorentz 3-Manifolds (2001)
Drumm, Todd A., Goldman, William M.
For isometric actions on flat Lorentz (2+1)-space whose linear part is a purely hyperbolic subgroup of O(2, 1), Margulis defined a marked signed Lorentzian length spectrum invariant closely related...
Flat Lorentz 3-Manifolds and Cocompact Fuchsian Groups (2000)
Goldman, William M., Margulis, Gregory A.
This paper gives a new proof of a result of Geoff Mess that the linear holonomy group of a complete flat Lorentz 3-manifold cannot be cocompact in SO(2,1). The proof uses a signed marked Lorentzian...
The Modular Group Action On Real SL(2)-Characters Of A PUNCTURED TORUS (2000)
. The group \Gamma of automorphisms of the polynomial (x; y; z) = x 2 + y 2 + z 2 \Gamma xyz \Gamma 2 is isomorphic to PGL(2; Z)n (Z=2 \Phi Z=2). We study the dynamics of the \Gamma-action on \Gamma1...
The topology of the relative character varieties of a quadruply-punctured sphere (1999)
Benedetto, Robert L., Goldman, William M.
Let M be a quadruply-punctured sphere with boundary components A,B,C,D. The $\Slt$-character variety of M consists of equivalence classes of homomorphisms $\rho$ of $\pi_1(M)\longrightarrow\Slt$ and...
Complex Hyperbolic Manifolds Homotopy Equivalent To A Riemann Surface (1998)
William M. Goldman, Michael Kapovich, Bernhard Leeb
. We construct actions of fundamental groups of Riemann surfaces by automorphisms of the complex hyperbolic plane, which realize all possible values of Toledo's invariant ø . For integer values...
The classification of real projective structures on compact surfaces (1997)
Choi, Suhyoung, Goldman, William M.
Choi gratefully acknowledges partial support from GARC-KOSEF. Goldman gratefully acknowledges partial support from the National Science Foundation, the Alfred P. Sloan Foundation and the Institute...
The Topology of the Relative Character Varieties of a Quadruply-Punctured Sphere (1997)
Robert L. Benedetto, William M. Goldman
. Let M be a quadruply-punctured sphere with boundary components A; B; C; D. The SL(2; C )-character variety of M consists of equivalence classes of homomorphisms ae of ß 1 (M ) \Gamma! SL(2; C )...
Complex Hyperbolic Manifolds Homotopy Equivalent To A Riemann Surface (1995)
A Riemann Surface, William M. Goldman, Michael Kapovich, Bernhard Leeb
. We construct isometric actions of fundamental groups of closed Riemann surfaces on the complex hyperbolic plane, which realize all possible values of Toledo's invariant . For integer values of...
Convex real projective structures on closed surfaces are closed (1993)
Choi, Suhyoung, Goldman, William M.
The first author's research was partially supported by a grant from TGRC-KOSEF and the second author's research was partially supported by University of Maryland Institute of Advanced Computer...
Book Review: An Extension Of Casson's Invariant By Kevin Walker (1993)
in terms of the fundamental group and reduces modulo two to Rohlin's invariant. In particular Rohlin's invariant cannot detect a counterexample to the Poincar'e conjecture. (For an...
Complete Flat Lorentzian 3-Manifolds with Free Fundamental Group (1990)
Todd A. Drumm, William M. Goldman
this paper we give a new proof of this result, using ideas from Margulis' original proof [3, 4] as well as geometric conditions due to the first author [1]. Let R
Projective geometry on manifolds Lecture Notes (v.0.3) - Mathematics 748B Spring 1988 (1988)
4.69> arises when one speaks only of points, lines and the relation of parallelism. And when one removes the notion of parallelism and only studies lines, points and the relation of incidence...
Abstract. Many interesting geometric structures on manifolds can be interpreted as structures locally modelled on homogeneous spaces. Given a homogeneous space (X, G) and a manifold M, there is a...
Flat Lorentz 3-manifolds and cocompact Fuchsian groups
William M. Goldman, Gregory A. Margulis
Introduction Consider Minkowski 2+1-space E and let G ae SO(2; 1) 0 be a discrete subgroup. Suppose that a group of affine isometries of E with linear part G acts properly and freely on E . In a...
Affine Schottky Groups and Crooked Tilings
Virginie Charette, William M. Goldman
This paper expounds these results.