Goldman, W., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200--225.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... all x K 1 g (e) H LG and Y ( 11, Proposition IV.G] v) for any x = x 1 , x 2g ) the kernel of # # x : LG equals the kernel of dK g (x) # : LG ker # # x = ker dK g (x) # = H # LG : Ad(x 1 )H = Ad(x 2g )H = H (32) 11, Proposition IV.C] and also in [9]) vi) If x Pfa#(# q(x) det # # x (33) where the Pfa#an is, as usual, the square root of the determinant of the matrix q(x) relative to an orthonormal basis ( 12, Proposition 3.3] vii) If f is a measurable function on K 1 g (e) invariant under the conjugation action ....

W. Goldman, The Symplectic Nature of Fundamental Groups of Surfaces, Adv. Math. 54, 200-225 (1984)

....on . Then # j # j g # for each j g , and hence ##, are relations on . The proof will involve the following lemma. 10.2) The only critical value of g is the identity matrix I . Proof. The derivative d g : g g at (A j , B j ) is easy to compute explicitly: see for example Goldman [8] or Gunning [10, Lemma 26] It is a sum of g terms, the kth being conjugate to (a j , b j ) AdA 1 k )b k AdB 1 k )a k . At a critical point, then, all g of these maps must fail to surject. For A #= I # G = SL(2, C) it is easy to check by hand, using Jordan canonical form, that the ....

W.M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984) 200--225.

....M fl whose strata are products of moduli of lower rank and whose index set consists of fl admissible partitions (Definition 4. 3) It is possible that this stratification coincides with the symplectic stratification of the representation variety of the fundamental group of surfaces given in [10] using the Mehta Seshadri theorem. We then prove a stratified version of Theorem 3.1 (see Theorem 4.5) When the singular moduli is defined by a weight on the boundary of the weight space, things become a little different. As before, the Jordan Holder filtration defines a natural stratification ....

W. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math., 54 (1984), 200--225.

....An exception is when r = 1, but then it is nothing but the complex torus T = C ) 2g . Instead of struggling with the singularities, choose any integer d coprime to r , and consider the space H = 1 (e 2 id=r I) G. This is a non compact complex manifold indeed, a smooth ane variety [14, 16] and will be our main object of study. It parametrizes gauge equivalence classes of G connections on C of constant central curvature di I , where is a 2 form on C chosen so that R C = 2 =r , and I is the r r identity matrix. Indeed, such a connection is again determined by its ....

W.M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984) 200-225.

....j j g for each j g , and hence , are relations on M g . The proof will involve the following lemma. 10.2) The only critical value of g is the identity matrix I . Proof. The derivative d g : g 2g g at (A j ; B j ) 2 G 2g is easy to compute explicitly: see for example Goldman [8] or Gunning [10, Lemma 26] It is a sum of g terms, the kth being conjugate to (a j ; b j ) 7 (I AdA 1 k )b k (I AdB 1 k )a k . At a critical point, then, all g of these maps must fail to surject. For A 6= I 2 G = SL(2; C ) it is easy to check by hand, using Jordan canonical form, that the ....

W.M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984) 200-225. 29

....theWeil Petersson 2 form is closed [Ah] We1] namely that theWeil Petersson metric is a K ahler metric on the complex manifold T (S) W.M. Goldman discovered that the Weil Petersson 2 form has a very topological interpretation, and can be expressed as a cup product in a twisted cohomology group [Go]. In this paper, we provide another topological expression for the Weil Petersson form, in terms of the shearing coordinates for Teichm uller space associated to a maximal geodesic lamination on the surface. More precisely, if we fix a maximal geodesic lamination #, the shearing coordinates for ....

....See Proposition 7 for a precise statement based on the conventions of 1. Since the Thurston form is constant in the shearing coordinates, it is closed as a di#erential form on H(#) A corollary of Theorem 1 is that it provides another proof that the Weil Petersson form is closed; compare [Ah] We1][Go]. In the special case where the maximal geodesic lamination # is obtained from a pair of pant decomposition of S by adding finitely many geodesics spiralling along these closed geodesics, the shearing coordinates are just linear combinations of the Fenchel Nielsen coordinates associated to this ....

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W.M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), 200--225.

....1 ( so(4; 1) PH 1 ( so(3; 1) Phi PH 1 ( R 4 1 ) When ae 0 is an isomorphism onto a non uniform lattice in SO 0 (3; 1) we have PH 1 ( so(4; 1) PH 1 ( R 4 1 ) The next results are standard consequences of duality for surfaces and three manifolds. Lemma 2.2. [8] Let be the fundamental group of a closed, orientable surface of genus g. Let G be a semisimple Lie group and fix a representation ae 0 : G. Then dimH 1 ( g) 2g Gamma 2) dimG 2 dimH 0 ( g) Proof. The Killing form on g is non degenerate and Ad invariant, and therefore gives ....

W. M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), no. 2, 200--225.

....g; h 2 , and a 1 coboundary is a 1 cocycle of the form c(g) 1 Gamma g)w for some w 2 V . Writing F n for the free group on n generators, we can make V into a ZF n module in a natural way. There is an isomorphism between V n and Z 1 (F n ; V ) given in terms of the Fox derivatives [7] (v 1 ; v n ) 7 (g 7 n X i=1 g x i v i ) From this one obtains Lemma 2.1. 7] Z 1 ( V ) f(v 1 ; v n ) 2 V n j n X i=1 r j x i v i = 0 for j = 1; pg Under this isomorphism, the subspace B 1 ( V ) of coboundaries consists of all elements of ....

....Writing F n for the free group on n generators, we can make V into a ZF n module in a natural way. There is an isomorphism between V n and Z 1 (F n ; V ) given in terms of the Fox derivatives [7] v 1 ; v n ) 7 (g 7 n X i=1 g x i v i ) From this one obtains Lemma 2.1. [7] Z 1 ( V ) f(v 1 ; v n ) 2 V n j n X i=1 r j x i v i = 0 for j = 1; pg Under this isomorphism, the subspace B 1 ( V ) of coboundaries consists of all elements of V n of the form ( 1 Gamma x 1 )w; 1 Gamma x 2 )w; 1 Gamma x n )w) for some ....

W. M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), no. 2, 200--225.

....invariants of the pseudo Anosov homeomorphism f : R R (like its Perron Frobenius matrix, see [FLP] Another interesting question is to describe stable and unstable manifolds of the fixed point [ of the mapping Phi. Note that Phi preserves the natural symplectic structure on X( 1 (R) see [Go]) Thus OE has 3g Gamma 3 eigenvalues whose absolute value is less than 1, and the same number of eigenvalues outside the unit disc (where g is the genus of R) In particular, the (complex) dimension of stable and unstable manifolds E s ; E u of Phi at [ is 3g Gamma 3. McMullen in ....

....projection X( 1 (R) Gamma of Hom( 1 (R) SL(2; C ) Gamma to X( 1 (R) is naturally isomorphic to the set theoretic quotient Hom( 1 (R) SL(2; C ) Gamma =SL(2; C ) see [JM, Theorem 1. 1] Moreover X( 1 (R) Gamma is a smooth complex manifold of the dimension 6g Gamma 6, see [W, Go]. Suppose that [ae 0 ] 2 X( 1 (R) is a fixed point for Phi and the image of ae 0 is Zariski dense in SL(2; C ) Then there is an element t 2 SL(2; C ) such that tae 0 (OE(fl) t Gamma1 = ae 0 (fl) for all fl 2 1 (R) We consider the induced action T 0 ( Phi) of Phi on the tangent ....

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W. Goldman, The symplectic nature of fundamental groups of surfaces, Advances in Math., Vol. 54 (1984) p. 200-- 225.

....character variety, moduli spaces, Diophantine Equations. 2 JOSEPH P. PREVITE AND EUGENE Z. XIA The space M C (SU(2) is compact, but possibly singular. The set of smooth points of M C (SU(2) possesses a natural symplectic structure which gives rise to a finite measure on M C (SU(2) see [2, 3, 5]) Let Diff(M; M) be the group of diffeomorphisms fixing M . The mapping class group Gamma is defined to be 0 (Diff(M; M ) The group Gamma acts on 1 (M) fixing the fl i s. It is known that (hence ) is invariant with respect to the Gamma action. In [2] Goldman showed that with ....

Goldman, W. M., The Symplectic Nature of Fundamental Groups of Surfaces, Adv. Math., Vol. 54, (1984), 200-225.

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Goldman, W., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200--225.

....n = 1. Then Hom(#, Z) # = H 1 (M ; Z n) acts on Hom(#, G) G by pointwise multiplication: if # # Hom(#, G) and u # Hom(#, Z) then u [#] # # # [#(#)u(#) The corresponding Hom(#, Z) action on functions t # is: u : t # # # t # # u 1 = u(#) 1 t # . 1) Recall the definition [1] of the symplectic structure on X. Suppose that # # Hom(#, G) is an irreducible representation. By Weil [6] compare also Raghunathan [5] the Zariski tangent space to Hom(#, G) at # identifies with the space Z 1 (#, g Ad# ) of cocycles where g Ad# is the # module defined by the ....

Goldman, W., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200--225.

....space of conjugacy classes of Fuchsian representations OE : Gamma SO(2; 1) 0 is an open subset of the space of conjugacy classes of all representations, which identifies with the Teichmuller space T(M) of M . See Weil [26, 27, 28] xVI of Raghunathan [22] for the general theory and Goldman [12, 13] for the case of surface groups. Its tangent space identifies with the cohomology group H 1 (G; R 2;1 ) where G = OE( Since the classical theory of Fuchsian groups is usually phrased in terms of SL(2; R) rather than SO(2; 1) and since 2 Theta2 matrices are more tractable than 3 Theta3 ....

Goldman, W., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200--225.

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W. Goldman, The Symplectic Nature of Fundamental Groups of Surfaces, Adv. Math. 54, 200-225 (1984)

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W.M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984) 200--225.

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