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- W108534959 abstract "The moduli space of (S, x) is the orbifold M(S, x) that classifies x-marked Riemann surfaces diffeomorphic to (S, x) up to isomorphism. Its universal cover (in the orbifold category) is the Teichmuller space T (S, x) that parametrizes complex structures on S up to isotopy relative to x. In other words, T (S, x) is the space of equivalence classes of oriented diffeomorphisms f : S → Σ of Riemann surfaces, where two oriented diffeomorphisms f : S → Σ and g : S → Σ of Riemann surfaces are equivalent if there exists a biholomorphic map h : Σ → Σ such that h ◦ f ≃ g are isotopic relative to x. In this context, the cover T (S, x) → M(S, x) is simply the forgetful map [f : S → Σ] 7→ [(Σ, f(x))]. It is also clear that the mapping class group Γ(S, x) = π0Diff+(S, x) of isotopy classes of oriented diffeomorphisms of (S, x) acts on T (S, x) by right composition and that M(S, x) identifies to the quotient T (S, x)/Γ(S, x). The group Γ(S, x) is infinite but it acts properly and discontinuously on T (S, x); in particular, the stabilizer of [f : S → Σ] is naturally isomorphic to the automorphism group Aut(Σ, f(x)), which is known to be finite (because χ(Ṡ) < 0). A classical result of Teichmuller states that T (S, x) is differentiably isomorphic to a Euclidean space (of dimension 6g − 6 + 2n). Thus, M(S, x) locally looks like C/G around [(Σ, f(x))], where G = Aut(Σ, x)." @default.
- W108534959 created "2016-06-24" @default.
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- W108534959 date "2008-10-01" @default.
- W108534959 modified "2023-09-27" @default.
- W108534959 title "Riemann surfaces, arc systems and Weil-Petersson form" @default.
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