William M. Goldman

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In [7] it was shown that if n is the fundamental group ofa 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,(More)
A symplectic structure on a manifold is a closed nondegenerate exterior 2form. The most common type of symplectic structure arises on a complex manifold as the imaginary part of a Hermitian metric which is Klhler. Many moduli spaces associated with Riemann surfaces have such Kahler structures: the Jacobi variety, Teichmiiller space, moduli spaces of stable(More)
where B : E x E —• E' is a bilinear map and E' is a vector space. (E may be identified with the Zariski tangent space to Q at 0.) Let V be an algebraic variety and x € V be a point. We say that V is quadratic at x if the analytic germ of V at x is equivalent to the germ of a quadratic cone at 0. Let T be a finitely generated group and G a k-algebraic group.(More)
Since rt is a finitely generated group, the space Hom0r, G) is a real analytic variety whenever G is a connected Lie group, and is a real affine algebraic variety whenever G is a linear algebraic group over R [3, 18, 27, 32]. The group G acts on Hom(Tr, G) by conjugation and the orbit space will be denoted by Horn(n, G)/G. Geometrically, the G-orbits on(More)
is isomorphic to PGL(2,Z)⋉ (Z/2⊕ Z/2). For t ∈ R , the Γ-action on κ(t) ∩ R displays rich and varied dynamics. The action of Γ preserves a Poisson structure defining a Γ–invariant area form on each κ(t) ∩ R . For t < 2, the action of Γ is properly discontinuous on the four contractible components of κ(t) ∩R and ergodic on the compact component (which is(More)
he deformation space t(E) of convex Rp2_ structures on a closed surface Y with X(z) < 0 is closed in the space Hom(7r, SL(3, IR))/SL(3, IR) of equivalence classes of representations r1 (l) -SL(3, IR) . Using this fact, we prove Hitchin's conjecture that the contractible "Teichmuller component" (Lie groups and Teichmuller space, preprint) of Hom(7r, SL(3,(More)
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 of 1 (M) ?! SL(2; C) and can be identiied with a quartic hypersurface in C 7. corresponding to representations with tr((A)) = a, tr((B)) = b, tr((C)) = c, tr((D)) = d is a cubic surface in C 3. We(More)