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William M. Goldman

Publications and source records attributed to William M. Goldman.

35 records · Page 2Linked to original sources
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↗
arXiv2009

Ergodicity of Mapping Class Group Actions on SU(2)-character varieties

Let S be a compact orientable surface with genus g and n boundary components d_1,...,d_n. Let b = (b_1, ..., b_n) where b_n lies in [-2,2]. Then the mapping class group of S acts on the relative SU(2)-character variety X comprising conjugacy classes of representations f of the fundamental group F of S, where trace f(d_i) = b_i. This action preserves a symplectic structure on the open dense smooth submanifold of X. corresponding to irreducible representations. This subset has full measure and is connected. In this note we use the symplectic geometry of this space to give a new proof that this action is ergodic.

math.GT↗
arXiv2009

Trace Coordinates on Fricke spaces of some simple hyperbolic surfaces

The conjugacy class of a generic unimodular 2 by 2 complex matrix is determined by its trace, which may be an arbitrary complex number. In the nineteenth century, it was known that a generic pair (X,Y) of such pairs is determined up to conjugacy by the triple of traces (tr(X),tr(Y),tr(XY), which may be an arbitary element of C^3. This paper gives an elementary and detailed proof of this fact, which was published by Vogt in 1889. The folk theorem describing the extension of a representation to a representation of the index-two supergroup which is a free product of three groups of order two, is described in detail, and related to hyperbolic geometry. When n > 2, the classification of conjugacy-classes of n-tuples in SL(2,C) is more complicated. We describe it in detail when n= 3. The deformation spaces of hyperbolic structures on some simple surfaces S whose fundamental group is free of rank two or three are computed in trace coordinates. (We only consider the two orientable surfaces whose fundamental group has rank 3.)

math.GT↗
arXiv2008

Higgs Bundles and Geometric Structures on Surfaces

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 Simpson. In particular we discuss the uniformization of Riemann surfaces by hyperbolic geometry from this viewpoint, and survey more recent developments in this theory.

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↗
arXiv2007

Notes on a paper of Mess

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 these notes in a forthcoming issue of Geometriae Dedicata.

math.DG↗
arXiv2006

Energy of Twisted Harmonic Maps of Riemann Surfaces

The energy of harmonic sections of flat bundles of nonpositively curved (NPC) length spaces over a Riemann surface $S$ is a function $E_ρ$ on Teichmüller space $\Teich$ which is a qualitative invariant of the holonomy representation $ρ$ of $π_1(S)$. Adapting ideas of Sacks-Uhlenbeck, Schoen-Yau and Tromba, we show that the energy function $E_ρ$ is proper for any convex cocompact representation of the fundamental group. More generally, if $ρ$ is a discrete embedding onto a normal subgroup of a convex cocompact group $Γ$, then $E_ρ$ defines a proper function on the quotient $\Teich/Q$ where $Q$ is the subgroup of the mapping class group defined by $Γ/ρ(π_1(S))$. When the image of $ρ$ contains parabolic elements, then $E_ρ$ is not proper. Using the recent solution of Marden's Tameness Conjecture, we show that if $ρ$ is a discrete embedding into $\SLtC$, then $E_ρ$ is proper if and only if $ρ$ is quasi-Fuchsian. These results are used to prove that the mapping class group acts properly on the subset of convex cocompact representations.

math.DG↗
arXiv2006

Mapping Class Group Dynamics on Surface Group Representations

Deformation spaces Hom($π$,G)/G of representations of the fundamental group $π$ of a surface $Σ$ in a Lie group $G$ admit natural actions of the mapping class group $Mod_Σ$, preserving a Poisson structure. When $G$ is compact, the actions are ergodic. In contrast if $G$ is noncompact semisimple, the associated deformation space contains open subsets containing the Fricke-Teichmüller space upon which $Mod_Σ$ acts properly. Properness of the $Mod_Σ$-action relates to (possibly singular) locally homogeneous geometric structures on $Σ$. We summarize known results and state open questions about these actions.

math.GT↗
arXiv2005

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

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 between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyper-Kaehlerstructure. The twistor space, real forms, and various group actions are computed explicitly in terms of the Jacobian of X. We describe the moduli spaces and their geometry in terms of the Riemann period matrix of X. This is the simplest case of the theory developed by Hitchin, Simpson and others. We emphasize its formal aspects that generalize to higher rank Higgs bundles over higher dimensional Kaehler manifolds.

math.DG↗
arXiv2005

An exposition of results of Fricke

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. Identities for triples of elements of SL(2,C) are also derived.

math.GM↗
arXiv2005

The Mapping Class Group acts reducibly on SU(n)-character varieties

When $G$ is a connected compact Lie group, and $π$ is a closed surface group, then $Hom(π,G)$ contains an open dense $Out(π)$-invariant subset which is a smooth symplectic manifold. This symplectic structure is $Out(π)$-invariant and therefore defines an invariant measure $μ$, which has finite volume. The corresponding unitary representation of $Out(π)$ on $L^2(Hom(π,G)/G,μ)$ contains no finite-dimensional subrepresentations besides the constants. This note gives a short proof that when $G=SU(n)$, the representation $L^2(Hom(π,G)/G,μ)$ contains many other invariant subspaces.

math.GT↗
arXiv2005

Is the deformation space of complete affine structures on the 2-torus smooth?

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 deformation space, under which it is diffeomorphic to $R^2$. The action of the mapping class group of $T^2$ is equivalent in these coordinates with the standard linear action of $\SL_2(Z)$ on $R^2$.

math.DG↗
arXiv2004

Homological action of the modular group on some cubic moduli spaces

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 on the subvarieties of its SL(2,C) character variety given by fixing the trace of the peripheral element (so-called "relative character varieties"). This mapping class group is isomorphic to PGL(2,Z). We also describe the corresponding mapping class group action for the four-holed sphere and its relative SL(2,C) character varieties, which are fibers of deformations x^2+y^2+z^2-xyz-2-Px-Qy-Rz of the above cubic. The 2-congruence subgroup PGL(2,Z)_(2) still acts on these cubics and is the full automorphism group when P,Q,R are distinct.

math.GT↗
arXiv2003

The complex-symplectic geometry of SL(2,C)-characters over surfaces

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 variety, we deduce that every G-invariant meromorphic function on X is constant. The trace functions of closed curves on M determine regular functions which generate complex Hamiltonian flows. For simple closed curves, these complex Hamiltonian flows arise from holomorphic flows on the representation variety generalizing the Fenchel-Nielsen twist flows on Teichmueller space and the complex quakebend flows on quasi-Fuchsian space. Closed curves in the complex trajectories of these flows lift to paths in the deformation space of complex-projective structures between different projective structures with the same holonomy (grafting). A pants decomposition determines a holomorphic completely integrable system on X. This integrable system is related to the complex Fenchel-Nielsen coordinates on quasi-Fuchsian space developed by Tan and Kourouniotis, and relate to recent formulas of Platis and Series on complex-length functions and complex twist flows.

math.DG↗
arXiv2003

The Modular Group Action on Real SL(2)-characters of a One-Holed Torus

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 = kappa^{-1}(t) intersect R displays rich and varied dynamics. The action of Gamma preserves a Poisson structure defining a Gamma-invariant area form on each ktR. For t < 2, the action of Gamma is properly discontinuous on the four contractible components of ktR and ergodic on the compact component (which is empty if t < -2). The contractible components correspond to Teichmueller spaces of (possibly singular) hyperbolic structures on a torus M-bar. For t = 2, the level set ktR consists of characters of reducible representations and comprises two ergodic components corresponding to actions of GL(2,Z) on (R/Z)^2 and R^2 respectively. For 2 < t <= 18, the action of Gamma on ktR is ergodic. Corresponding to the Fricke space of a three-holed sphere is a Gamma-invariant open subset Omega subset R^3 whose components are permuted freely by a subgroup of index 6 in Gamma. The level set ktR intersects Omega if and only if t > 18, in which case the Gamma-action on the complement ktR - Omega is ergodic.

math.DG↗
arXiv2000

Flat Lorentz 3-Manifolds and Cocompact Fuchsian Groups

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 length-spectrum invariant developed by G.Margulis, reinterpreted in terms of deformations of hyperbolic surfaces.

math.DG↗