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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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We work over a fixed field k, either the real or complex numbers. A quadratic cone in a k-vector space E is an affine variety Q C E defined by equations B(u,u) = 0where 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. We identify G with its set of k-points, which has the natural structure of a Lie group over k. Then the set Hom(r, G) of all homomorphisms r -• G equals the set of k-points of a k-algebraic variety 9t(r,G) (compare [9,10] Suppose that {H p > q , Q} is a polarized Hodge structure of weight n, that G is the group of real points of the isometry group of Q and X = G/V is the classifying space for polarized Hodge structures of the above type (see Griffiths [8, p. 15]). Suppose further M is a complex manifold with fundamental group T. A representation p: T -• G determines a flat principal G-bundle P p over M and an associated X-bundle P p XQ X. A horizontal holomorphic Vreduction of P p is a holomorphic section of P p XQ X whose differential carries the holomorphic tangent bundle of M into the horizontal subbundle Th(X) defined in [8, p. 20]. THEOREM 2. Let M be a compact Kàhler manifold with fundamental group T and X = G/V a classifying space for polarized Hodge structures. Suppose that p: T -• G is a representation such that the associated principal bundle over M admits a horizontal holomorphic V-reduction. Then 9l(r, G) 'is quadratic at p.

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Those groups r which act properly discontinuously and aillnely on II?' with compact fundamental domain are classified. First it is shown that such a group f contains a solvable subgroup of finite index, thus establishing a conjecture of Auslander in dimension three. Then unimodular simply transitive alTine actions on IR' are classified; this leads to a classification of atTine crystallographic groups acting on IR3. A characterization of which abstract groups admit such an action is given; moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups (respectively simply transitive afline groups) is induced by conjugation by a polynomial automorphism of the affrne space. A characterization is given of which closed 3-manifolds can be represented as quotients of IR' by groups of afftne transformations: a closed 3-manifold M admits a complete atline structure if and only if M has a finite covering homeomorphic (or homotopy-equivaient) to a 2-torus bundle over the circle.

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The group Γ of automorphisms of the polynomial κ(x, y, z) = x 2 + y 2 + z 2 − xyz − 2 is isomorphic to PGL(2, Z) ⋉ (Z/2 ⊕ Z/2).For t ∈ R, the Γ-action on κ −1 (t) ∩ R 3 displays rich and varied dynamics. The action of Γ preserves a Poisson structure defining a Γ-invariant area form on each κ −1 (t) ∩ R 3 . For t < 2, the action of Γ is properly discontinuous on the four contractible components of κ −1 (t) ∩ R 3 and ergodic on the compact component (which is empty if t < −2). The contractible components correspond to Teichmüller spaces of (possibly singular) hyperbolic structures on a torusM . For t = 2, the level set κ −1 (t) ∩ R 3 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 Γ on κ −1 (t) ∩ R 3 is ergodic. Corresponding to the Fricke space of a three-holed sphere is a Γ-invariant open subset Ω ⊂ R 3 whose components are permuted freely by a subgroup of index 6 in Γ. The level set κ −1 (t) ∩ R 3 intersects Ω if and only if t > 18, in which case the Γ-action on the complement (κ −1 (t) ∩ R 3 )−Ω is ergodic.