Meromorphic Projective Structures: Signed Spaces, Grafting and Monodromy

TL;DR

The paper develops a comprehensive framework for meromorphic projective structures on marked bordered surfaces with signed regular singularities. It extends Thurston’s grafting theory to signed spaces, proving a signed grafting map $ ext{widehat{Gr}}$ is a homeomorphism between $ ext{T}^{ ext{±}}(bS,bM) imes ext{ML}^{ ext{±}}(bS,bM)$ and $ ext{P}^{ ext{±}}(bS,bM)$, with a fiber-product description via $c_p=l+i\alpha$ for interior punctures. It then defines a framed monodromy map $ ext{widehat{ ext{$ extPhi$}}}$ from signed projective structures to framed PSL$_2(bC)$ representations, extends the framing to regular singularities via asymptotic data, and characterizes its image as the non-degenerate framed representations; the map is shown to be a local biholomorphism, echoing Ehresmann–Thurston-type results. Together, these results yield a robust, interoperable picture linking grafting, Schwarzian data, monodromy, and framed representation theory, and they extend known results from the irregular-only setting to the broad meromorphic context with apparent singularities and signs at punctures.

Abstract

A meromorphic quadratic differential on a compact Riemann surface defines a complex projective structure away from the poles via the Schwarzian equation. In this article we first prove the analogue of Thurston's Grafting Theorem for the space of such structures with signings at regular singularities. This extends previous work of Gupta-Mj which only considered irregular singularities. We also define a framed monodromy map from the signed space extending work of Allegretti-Bridgeland, and we characterize the PSL(2,C)-representations that arise as holonomy, generalizing results of Gupta-Mj and Faraco-Gupta. As an application of our Grafting Theorem, we also show that the monodromy map to the moduli space of framed representations (as introduced by Fock-Goncharov) is a local biholomorphism, proving a conjectured analogue of a result of Hejhal.

Figures (6)

• Figure 1: Fiber product defining the space $\mathcal{P}^{\pm}(\mathbb{S},\mathbb{M})$. Here, $\pi$ is the forgetful projection map.
• Figure 3: Grafting description of the projective structures in Example 1 (left) and Example 2 (right). The blue geodesics have weight $\pi$ and the red geodesic has weight $2\pi$.
• Figure 4: Lifts of a spiralling leaf to the universal cover $\mathbb{H}^2$ accumulates to a lift of the geodesic boundary component. The compact set $K$ (shown shaded) intersects finitely many of such lifts.
• Figure 5: A hyperbolic surface $Z$ as in Lemma \ref{['lem:immT']}. One can choose three arcs from the marked point $p_0$ (at the cusp end) to itself that wind around the geodesic boundary component and bound an immersed ideal triangle.
• Figure 6: A possible hyperbolic surface $X$ and geodesic lamination $\lambda^\prime$: the metric completion of $X \setminus \lambda^\prime$ is shown in Figure 5.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Definition 2.1
• Definition 2.2: Definition 3.4 of All
• Theorem 2.3: Proposition 3.5. of All
• Definition 2.4
• Lemma 2.5