Branched projective structures with Fuchsian holonomy

Abstract

We prove that if S is a closed compact surface of genus g ≥ 2, and if ρ : π 1 (S) → PSL(2, C) is a quasi-Fuchsian representation, then the deformation space M k,ρ of branched projective structures on S with total branching order k and holonomy ρ is connected, as soon as k > 0. Equivalently, two branched projective structures with the same quasi-Fuchsian holonomy and the same number of branch points are related by a movement of branch points. In particular grafting annuli are obtained by moving branch points. In the appendix we give an explicit atlas for M k,ρ for non-elementary representations ρ. It is shown to be a smooth complex manifold modeled on Hurwitz spaces.

Mapping class group dynamics and the holonomy of branched affine structures

“…The class up to conjugation of the holonomy representation therefore defines a geometric invariant of the structure. This invariant is known to be far from classifying; describing the set of surfaces having the same holonomy as been investigated in several contexts (see for instance [1] or [4] in the case of branched projective structures).…”

Stieltjes–Bethe equations in higher genus and branched coverings with even ramifications

“…The previous construction gives examples of so-called branched projective structures [16,17]. For recent developments in this area we refer to [2,8]. The branched projective structures on a Riemann surface can be characterized in terms of equations (1.1) with meromorphic potential in the following way: all P SL(2) monodromies of the developing map around singularities of U are trivial.…”

“…• Monodromy groups (in any genus) corresponding to so-called "branched projective structures" introduced in [16,17] and studied more recently in [2]. Branched projective structures are defined by the condition that all SL(2) monodromies of equation (1.1) corresponding to generators γ i , i = 1, .…”

The oriented graph of multi-graftings in the Fuchsian case

“…We denote by P (ρ) the set of equivalence classes of marked CP 1 -structures on a surface of genus g with holonomy ρ, where two projective structures ( S i , D i ), i = 1, 2 are equivalent if there exists a Γ g -equivariant diffeomorphism Φ : S 1 → S 2 such that D 1 = D 2 • Φ. This definition of projective structure coincides with the classical one because there is no ambiguity in the choice of developing map when the holonomy representation is non-elementary, see [2,Lemma 12.10].…”

“…Grafting was used by Hejhal [5,Theorem 4] and Thurston (unpublished) to produce examples of projective structures with holonomy ρ that are different from the uniformizing structure σ u = ρ(Γ g )\H 2 . Such structures are called exotic.…”