The monodromy groups of Schwarzian equations on closed Riemann surfaces. (English) Zbl 0977.30028

Gallo, Daniel; Kapovich, Michael; Marden, Albert

Ann. Math. (2) 151, No. 2, 625-704 (2000).

This great paper presents a complete, self-contained proof of the following results: Theorem: Let \(R\) be an oriented closed surface of genus exceeding one, and \[ \theta: \pi_1(R;0) \to\Gamma \subset PSL(2, \mathbb{C}) \] a homomorphism of its fundamental group onto a nonelementary group \(\Gamma\) of Möbius transformations. Then: (i) \(\theta\) is reduced by a complex projective structure for some complex structure on \(R\) if and only if \(\theta\) lifts to a homomorphism \[ \theta: \pi_1 (R,0)\to SL(2,\mathbb{C}) \] (ii) \(\theta\) is induced by a branched complex projective structure with a single branch point of order two for some complex structure on \(R\) if and only if \(\theta\) does not lift to a homomorphism into \(SL(2;\mathbb{C})\).
Editorial remark (2023): In [T. Le Fils, Ann. Inst. Fourier 73, No. 1, 423–445 (2023; Zbl 07688659)] the author asserts that there is a gap in a proof in the present paper and gives a correct proof.

Reviewer: Lixin Liu (Guangzhou)

MSC:

30F99	Riemann surfaces
57M50	General geometric structures on low-dimensional manifolds
30F10	Compact Riemann surfaces and uniformization
30F40	Kleinian groups (aspects of compact Riemann surfaces and uniformization)
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Citations:

Zbl 07688659