Handbook of Teichmüller Theory, Volume II
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We characterize topological monodromies of holomorphic families of Riemann surfaces over the punctured disk. A holomorphic family (M, π, Δ*) of type (g, n) over Δ* is a triplet of a two-dimensional complex manifold M, the punctured disk Δ* and a holomorphic map π : M → Δ* such that for every t ∈ Δ* the fiber St = π−1(t) is a Riemann surface of fixed finite type (g, n) with 2g − 2 + n > 0 and the complex structure of St depends holomorphically on the parameter t. The topological monodromy of (M, π, Δ*) around the origin is given by an element [ω] of the mapping class group Mg, n(Σ) of an oriented topological surface Σ of type (g, n). Our main result sates that an element [ω] of Mg, n(Σ) is a topological monodromy of a holomorphic family (M, π, Δ*) if and only if ω : Σ → Σ is a pseudo-periodic map of negative type.
Keywords: Holomorphic family, Thurston–Bers classification, topological monodromy, rigidity, Riemann surface with nodes