2017 • 142 Pages • 1.09 MB • English

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Generalized Teichmu¨ller Spaces Anton Zeitlin Outline Introduction Generalized Teichmu¨ller Spaces, Spin Structures and Cast of characters Ptolemy Transformations Coordinates on Super-Teichmu¨ller space N = 2 Super-Teichmu¨ller Anton M. Zeitlin theory Open problems Columbia University, Department of Mathematics Louisiana State University Baton Rouge February, 2017

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Generalized Outline Teichmu¨ller Spaces Anton Zeitlin Outline Introduction Introduction Cast of characters Coordinates on Super-Teichmu¨ller space N = 2 Cast of characters Super-Teichmu¨ller theory Open problems Coordinates on Super-Teichmu¨ller space N = 2 Super-Teichmu¨ller theory Open problems

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Generalized Introduction Teichmu¨ller Spaces Anton Zeitlin Outline g Introduction Let Fs ≡ F be the Riemann surface of genus g and s punctures. Cast of characters We assume s > 0 and 2 − 2g − s < 0. Coordinates on Super-Teichmu¨ller space N = 2 Super-Teichmu¨ller theory Open problems Teichmu¨ller space T(F) has many incarnations: ◮ {complex structures on F}/isotopy ◮ {conformal structures on F}/isotopy ◮ {hyperbolic structures on F}/isotopy Isotopy here stands for diﬀeomorphisms isotopic to identity.

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Generalized Introduction Teichmu¨ller Spaces Anton Zeitlin Outline g Introduction Let Fs ≡ F be the Riemann surface of genus g and s punctures. Cast of characters We assume s > 0 and 2 − 2g − s < 0. Coordinates on Super-Teichmu¨ller space N = 2 Super-Teichmu¨ller theory Open problems Teichmu¨ller space T(F) has many incarnations: ◮ {complex structures on F}/isotopy ◮ {conformal structures on F}/isotopy ◮ {hyperbolic structures on F}/isotopy Isotopy here stands for diﬀeomorphisms isotopic to identity.

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Generalized Introduction Teichmu¨ller Spaces Anton Zeitlin Outline g Introduction Let Fs ≡ F be the Riemann surface of genus g and s punctures. Cast of characters We assume s > 0 and 2 − 2g − s < 0. Coordinates on Super-Teichmu¨ller space N = 2 Super-Teichmu¨ller theory Open problems Teichmu¨ller space T(F) has many incarnations: ◮ {complex structures on F}/isotopy ◮ {conformal structures on F}/isotopy ◮ {hyperbolic structures on F}/isotopy Isotopy here stands for diﬀeomorphisms isotopic to identity.

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Generalized Teichmu¨ller Spaces Anton Zeitlin Representation-theoretic deﬁnition: Outline ′ Introduction T(F) = Hom (π1(F), PSL(2, R))/PSL(2, R), Cast of characters ′ Coordinates on where Hom stands for Homs such that the group elements Super-Teichmu¨ller space corresponding to loops around punctures are parabolic (|tr| = 2). N = 2 Super-Teichmu¨ller theory The image Γ ∈ PSL(2, R) is a Fuchsian group. Open problems + By Poincar´e uniformization we have F = H /Γ, where PSL(2, R) acts + on the hyperbolic upper-half plane H as oriented isometries, given by fractional-linear transformations: az + b z → . cz + d ˜ + + The punctures of F = H belong to the real line ∂H , which is called absolute.

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Generalized Teichmu¨ller Spaces Anton Zeitlin Representation-theoretic deﬁnition: Outline ′ Introduction T(F) = Hom (π1(F), PSL(2, R))/PSL(2, R), Cast of characters ′ Coordinates on where Hom stands for Homs such that the group elements Super-Teichmu¨ller space corresponding to loops around punctures are parabolic (|tr| = 2). N = 2 Super-Teichmu¨ller theory The image Γ ∈ PSL(2, R) is a Fuchsian group. Open problems + By Poincar´e uniformization we have F = H /Γ, where PSL(2, R) acts + on the hyperbolic upper-half plane H as oriented isometries, given by fractional-linear transformations: az + b z → . cz + d ˜ + + The punctures of F = H belong to the real line ∂H , which is called absolute.

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Generalized Teichmu¨ller Spaces Anton Zeitlin Representation-theoretic deﬁnition: Outline ′ Introduction T(F) = Hom (π1(F), PSL(2, R))/PSL(2, R), Cast of characters ′ Coordinates on where Hom stands for Homs such that the group elements Super-Teichmu¨ller space corresponding to loops around punctures are parabolic (|tr| = 2). N = 2 Super-Teichmu¨ller theory The image Γ ∈ PSL(2, R) is a Fuchsian group. Open problems + By Poincar´e uniformization we have F = H /Γ, where PSL(2, R) acts + on the hyperbolic upper-half plane H as oriented isometries, given by fractional-linear transformations: az + b z → . cz + d ˜ + + The punctures of F = H belong to the real line ∂H , which is called absolute.

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Generalized Teichmu¨ller Spaces Anton Zeitlin Representation-theoretic deﬁnition: Outline ′ Introduction T(F) = Hom (π1(F), PSL(2, R))/PSL(2, R), Cast of characters ′ Coordinates on where Hom stands for Homs such that the group elements Super-Teichmu¨ller space corresponding to loops around punctures are parabolic (|tr| = 2). N = 2 Super-Teichmu¨ller theory The image Γ ∈ PSL(2, R) is a Fuchsian group. Open problems + By Poincar´e uniformization we have F = H /Γ, where PSL(2, R) acts + on the hyperbolic upper-half plane H as oriented isometries, given by fractional-linear transformations: az + b z → . cz + d ˜ + + The punctures of F = H belong to the real line ∂H , which is called absolute.

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Generalized Teichmu¨ller Spaces Anton Zeitlin Representation-theoretic deﬁnition: Outline ′ Introduction T(F) = Hom (π1(F), PSL(2, R))/PSL(2, R), Cast of characters ′ Coordinates on where Hom stands for Homs such that the group elements Super-Teichmu¨ller space corresponding to loops around punctures are parabolic (|tr| = 2). N = 2 Super-Teichmu¨ller theory The image Γ ∈ PSL(2, R) is a Fuchsian group. Open problems + By Poincar´e uniformization we have F = H /Γ, where PSL(2, R) acts + on the hyperbolic upper-half plane H as oriented isometries, given by fractional-linear transformations: az + b z → . cz + d ˜ + + The punctures of F = H belong to the real line ∂H , which is called absolute.

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