The Symplectic Nature of Fundamental Groups of Surfaces

  title={The Symplectic Nature of Fundamental Groups of Surfaces},
  author={William M. Goldman},
  journal={Advances in Mathematics},
  • W. Goldman
  • Published 1 November 1984
  • Mathematics
  • Advances in Mathematics

Invariant functions on Lie groups and Hamiltonian flows of surface group representations

Si π est le groupe fondamental d'une surface orientee fermee S et G est un groupe de Lie satisfaisant des conditions tres generales, alors l'espace Hom (π,G)/G des classes de conjugaison de

A note on exceptional groups and Reidemeister torsion

Let Σ be a closed orientable surface of genus at least 2 and G be one of the exceptional groups G2, F4, and E6. The present article considers the set Rep(Σ, G) of G-valued representations from the

Action of the Johnson-Torelli group on representation varieties

Let Σ be a compact orientable surface with genus g and n boundary components B = (B1, . . . , Bn). Let c = (c1, . . . , cn) ∈ [−2, 2]n. Then the mapping class group MCG of Σ acts on the relative

A Foliation of the Space of Conjugacy Classes of Representations of a Surface Group

Let Π be the fundamental group of a closed orientable surface of genus g ≥ 1, and let R(Π, G)/G be the space of conjugacy classes of representations of Π into a connected real reductive Lie group G.

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


Deformation spaces Hom(�,G)/G of representations of the fundamental groupof a surfacein a Lie group G ad- mit natural actions of the mapping class group Mod�, preserving a Poisson structure. When G


The representation space X(G) = Hom(π, G)/G of the fundamental group π of a Riemann surface Σ of genus g ≥ 2 is the symplectic reduction of the extended moduli space defined in [6]. Using this

Non-injective representations of a closed surface group into PSL ( 2 , R ) By

Let e denote the Euler class on the space Hom(Γg, P SL(2, R)) of representations of the fundamental group Γg of the closed surface Σg. Goldman showed that the connected components of Hom(Γg, P SL(2,

Covering spaces of character varieties Sean Lawton and

Let Γ be a finitely generated discrete group. Given a covering map H → G of Lie groups with G either compact or complex reductive, there is an induced covering map Hom(Γ, H) → Hom(Γ, G). We show that

Gluing Formulas for Volume Forms on Representation Varieties of Surfaces

. Let Σ g,n be a compact oriented surface with genus g ≥ 2 bordered by n circles. Due to Witten, the twisted Reidemeister torsion coincides with a power of the Atiyah-Bott-Goldman-Narasimhan



Characteristic classes and representations of discrete subgroups of Lie groups

A volume invariant is used to characterize those representations of a countable group into a connected semisimple Lie group G which are injective and whose image is a discrete cocompact subgroup of

Deformations of homomorphisms of Lie groups and Lie algebras

The purpose of this note is to announce several results on deformations of homomorphisms of Lie groups and Lie algebras. Our main theorems are precise analogues of two basic theorems on deformations

Braids, Links, and Mapping Class Groups.

The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned with

Stable and unitary vector bundles on a compact Riemann surface

Let X be a compact Riemann surface of genus g _ 2. A holomorphic vector bundle on X is said to be unitary if it arises from a unitary representation of the fundamental group of X. We prove in this

The Fenchel-Nielsen deformation

The uniformization theorem provides that a Riemann surface S of negative Euler characteristic has a metric of constant curvature -1. A hyperbolic structure can be understood in terms of its

On the symplectic geometry of deformations of a hyperbolic surface

Let R be a Riemann surface. In this manuscript we consider a geometry on the moduli space X(R) for R, which we regard as the space of equivalence classes of constant curvature metrics on the

Discrete subgroups of Lie groups

Preliminaries.- I. Generalities on Lattices.- II. Lattices in Nilpotent Lie Groups.- III. Lattices in Solvable Lie Groups.- IV. Polycyclic Groups and Arithmeticity of Lattices in Solvable Lie

Lectures on Symplectic Manifolds

Introduction Symplectic manifolds and lagrangian submanifolds, examples Lagrangian splittings, real and complex polarizations, Kahler manifolds Reduction, the calculus of canonical relations,

The Yang-Mills equations over Riemann surfaces

  • M. AtiyahR. Bott
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1983
The Yang-Mills functional over a Riemann surface is studied from the point of view of Morse theory. The main result is that this is a ‘perfect' functional provided due account is taken of its gauge