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Branched \(\mathbb{C} P^ 1\)-structures on surfaces with prescribed real holonomy. (English) Zbl 0833.53054
Due to a result of Gallo-Goldman-Porter (preprint), a representation \(\in \operatorname{Hom} (\pi, \text{PSL} (2, \mathbb{R}))\) of the fundamental group \(\pi\) of a closed oriented surface \(F_g\) of genus \(g \geq 2\) occurs as the holonomy representation of a \(\mathbb{C} P^1\)-structure on \(F_g\) if and only if its Euler class \(e(\rho)= 0\pmod 2\) (i.e., \(\rho\) admits a lift to \(\text{SL} (2, \mathbb{R})\)), and \(\rho (\pi)\) is not an elementary subgroup of \(\text{PSL} (2, \mathbb{R})\).
The goal of the present paper is to show that representations \(\rho\in \operatorname{Hom} (\pi, \text{PSL} (2, \mathbb{R}))\) with \(e(\rho)= 1\pmod 2\) nonetheless occur as the holonomy representations of branched \(\mathbb{C} P^1\)- structures with one branch point of degree 2. See also W. M. Goldman [Invent. Math. 93, No. 3, 557-607 (1988; Zbl 0655.57019)] and D. M. Gallo [Bull. Am. Math. Soc., New. Ser. 20, No. 1, 31-34 (1989; Zbl 0674.30032)].

53C65 Integral geometry
57M05 Fundamental group, presentations, free differential calculus
30F10 Compact Riemann surfaces and uniformization
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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