The symplectic nature of fundamental groups of surfaces

Goldman, William M.

doi:10.1016/0001-8708(84)90040-9

Cited by 548 publications

(547 citation statements)

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“…The next ingredients are the spaces Hom(n, G)jG with their symplectic structure defined by m; for details we refer the reader to Goldman [7]. Recall that Hom(n, G) denotes the real analytic variety of all homomorphisms n~G and Hom(n, G)jG is its quotient by the action of G on Hom(n, G) by inner automorphisms.…”

Section: T T=omentioning

confidence: 99%

“…Recall that Hom(n, G) denotes the real analytic variety of all homomorphisms n~G and Hom(n, G)jG is its quotient by the action of G on Hom(n, G) by inner automorphisms. As shown in [7], the singular subset of Hom(n, G) consists of representations ¢J E Hom(n, G) such that the centralizer of¢J(n) in Ad G has positive dimension; moreover G acts locally freely on the set of smooth points Hom(n, G) . After removing possibly more G-invariant subsets of large codimension, one obtains a Zariski-open subset DC Hom(n, G) such that DjG is a Hausdorff smooth manifold.…”

Section: T T=omentioning

confidence: 99%

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

1986

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In [7] it was shown that if n is the fundamental group of a closed oriented surface S and G is Lie group satisfying very general conditions, then the space Hom(n, G)/G of conjugacy classes of representation n-+G has a natural symplectic structure. This symplectic structure generalizes the Weil-Petersson Kahler form on Teichmiiller space (taking G= PSL(2, lR)), the cup-product linear symplectic structure on H 1 (S, lR) (when G= lR), and the Kahler forms on the Jacobi variety of a Riemann surface M homeomorphic to S (when G= U(l)) and the NarasimhanSeshadri moduli space of semistable vector bundles of rank n and degree 0 on M (when G = U(n)). The purpose of this paper is to investigate the geometry of this symplectic structure with the aid of a natural family of functions on Hom(n, G)/G.The inspiration for this paper is the recent work of Scott Wolpert on the WeilPetersson symplectic geometry of Teichmiiller space [18][19][20]. In particular he showed that the Fenchel-Nielsen "twist flows" on Teichmiiller space are Hamiltonian flows (with respect to the Weil-Petersson Kahler form) whose associated potential functions are the geodesic length functions. Moreover he found striking formulas which underscore an intimate relationship between the symplectic geometry of Teichmiiller space and the hyperbolic geometry (and hence the topology) of the surface. In particular the symplectic product of two twist vector fields (the Poisson bracket of two geodesic length functions) is interpreted in terms of the geometry of the surface.We will reprove these formulas of Wolpert in our more general context. Accordingly our proofs are simpler and not restricted to Teichmiiller space: while Wolpert's original proofs use much of the machinery of Teichmiiller space theory, we give topological proofs which involve the multiplicative properties of homology with local coefficients and elementary properties of invariant functions.Before stating the main results, it will be necessary to describe the ingredients of the symplectic geometry. Let G be a Lie group with Lie algebra g. The basic property we need concerning G is the existence of an orthogonal structure on G: an orthogonal structure on G is a nondegenerate symmetric bilinear form~: 9 x g-+ lR *

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Section: T T=omentioning

confidence: 99%

Section: T T=omentioning

confidence: 99%

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

1986

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“…We will denote the Gcharacter variety of N by X (N, G) or, when G is understood to be PSL 2 C, by X(N ). Recall that there is a natural holomorphic structure that X(N ) inherits from PSL 2 C. See [Gol1] and [Gol2] for details.…”

Section: Background and Notationmentioning

confidence: 99%

“…One interpretation of this structure identifies T [ρ] X(Σ) with H 1 (Σ, (sl 2 C) ρ ), the vector space of 1-forms with values in the flat sl 2 C-bundle on Σ associated to ρ. (See [Gol1] for details about the infinitesimal deformation theory of X(Σ)). The restriction map p * : X(Σ) → X(Σ ) naturally induces the pullback map p * :…”

Section: Skinning Maps Of Finite Covers Of Mmentioning

confidence: 99%

A family of non-injective skinning maps with critical points

Gaster

2015

Trans. Amer. Math. Soc.

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Abstract. Certain classes of 3-manifolds, following Thurston, give rise to a 'skinning map', a self-map of the Teichmüller space of the boundary. This paper examines the skinning map of a 3-manifold M , a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of M to conclude that the skinning map associated to M sends a specified path to itself, and use estimates on extremal length functions to show non-monotonicity and the existence of a critical point. A family of finite covers of M produces examples of non-immersion skinning maps on the Teichmüller spaces of surfaces in each even genus, and with either 4 or 6 punctures.

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