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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)].

##### MSC:
 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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