TAILIEUCHUNG - Đề tài " Dynamics of SL2(R) over moduli space in genus two "

Annals of Mathematics By Curtis T. McMullen* .Annals of Mathematics, 165 (2007), 397–456 Dynamics of SL2(R) over moduli space in genus two By Curtis T. McMullen* Abstract This paper classifies orbit closures and invariant measures for the natural action of SL2 (R) on ΩM2 , the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two. Contents 1. Introduction 2. Dynamics and Lie groups 3. Riemann surfaces and holomorphic 1-forms 4. Abelian varieties with real multiplication 5. Recognizing eigenforms 6. Algebraic sums of 1-forms 7. Connected sums of 1-forms 8. Eigenforms as connected sums 9. Pairs of splittings 10. Dynamics on. | Annals of Mathematics Dynamics of SL2 R over moduli space in genus two By Curtis T. McMullen Annals of Mathematics 165 2007 397 456 Dynamics of SL2 R over moduli space in genus two By Curtis T. McMullen Abstract This paper classifies orbit closures and invariant measures for the natural action of SL2 R on dM2. the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two. Contents 1. Introduction 2. Dynamics and Lie groups 3. Riemann surfaces and holomorphic 1-forms 4. Abelian varieties with real multiplication 5. Recognizing eigenforms 6. Algebraic sums of 1-forms 7. Connected sums of 1-forms 8. Eigenforms as connected sums 9. Pairs of splittings 10. Dynamics on QM2 2 11. Dynamics on QM2 1 1 12. Dynamics on QEp 1. Introduction Let Mg denote the moduli space of Riemann surfaces of genus g. By Teichmuller theory every holomorphic 1-form ư z dz on a surface X E Mg generates a complex geodesic f H2 Mg isometrically immersed for the Teichmuuller metric. Research partially supported by the NSF. 398 CURTIS T. MCMULLEN In this paper we will show Theorem . Let f H2 M2 be a complex geodesic generated by a holomorphic 1-form. Then f H2 is either an isometrically immersed algebraic curve a Hilbert modular surface or the full space M2. In particular f H2 is always an algebraic subvariety of M2. Raghunathan s conjectures. For comparison consider a finite volume hyperbolic manifold M in place of Mg. While the closure of a geodesic line in M can be rather wild the closure of a geodesic plane f H2 M Hra T is always an immersed submanifold. Indeed the image of f can be lifted to an orbit of U SL2 R on the frame bundle FM G r G SO n 1 . Raghunathan s conjectures proved by Ratner then imply that Ux Hx c G r for some closed subgroup H c G meeting xTx-1 in a lattice. Projecting back to M one finds that f H2 c M is an immersed hyperbolic fc-manifold with 2 k n Sh . The study of complex geodesics in Mg is similarly related to the dynamics of SL2 R on the .

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