[Paper Review] Holonomy limits of complex projective structures

David Dumas|arXiv (Cornell University)|May 25, 2011

Geometric and Algebraic Topology30 references4 citations

TL;DR

This paper investigates the asymptotic behavior of holonomy representations associated with complex projective structures on compact Riemann surfaces, showing that the Morgan-Shalen boundary limits of these representations are governed by the R-trees dual to the horizontal foliations of quadratic differentials. The key result establishes that for differentials with only simple zeros, the holonomy limit is uniquely represented by the dual R-tree, while for abelian limits, the underlying differential must be the square of a holomorphic 1-form with harmonic representative in the corresponding cohomology class.

ABSTRACT

We study the limits of holonomy representations of complex projective structures on a compact Riemann surface in the Morgan-Shalen compactification of the character variety. We show that the dual R-trees of the quadratic differentials associated to a divergent sequence of projective structures determine the Morgan-Shalen limit points up to a natural folding operation. For quadratic differentials with simple zeros, no folding is possible and the limit of holonomy representations is isometric to the dual tree. We also derive an estimate for the growth rate of the holonomy map in terms of a norm on the space of quadratic differentials.

Motivation & Objective

• To understand the large-scale behavior of the holonomy map from the space of quadratic differentials to the SL(2,C) character variety.
• To characterize the accumulation points of holonomy representations in the Morgan-Shalen compactification of the character variety.
• To determine the relationship between the geometry of quadratic differentials and the limiting actions on R-trees.
• To establish conditions under which the limiting R-tree is uniquely determined by the projective limit of the quadratic differential.

Proposed method

• Uses the Morgan-Shalen compactification to analyze boundary points as projective equivalence classes of length functions on the fundamental group.
• Constructs equivariant, straight maps from the R-tree dual to a quadratic differential to limiting R-trees representing holonomy limits.
• Applies the theory of equivariant maps and Busemann functions to analyze abelian length functions and their associated actions.
• Employs a visual extension procedure to relate surfaces in hyperbolic 3-space to the holonomy representations.
• Utilizes the scale of translation lengths in hyperbolic 3-space, showing they grow as ||φ||^{1/2} for a quadratic differential φ.
• Applies results from Culler and Morgan on uniqueness of R-tree realizations of length functions, excluding abelian cases.

Experimental results

Research questions

• RQ1 What is the limiting behavior of holonomy representations as the norm of a quadratic differential diverges?
• RQ2 How do the R-trees dual to the horizontal foliations of quadratic differentials relate to the Morgan-Shalen boundary limits of holonomy representations?
• RQ3 Under what conditions is the limiting R-tree action on an R-tree uniquely determined by the projective limit of the quadratic differential?
• RQ4 Can abelian length functions arise as limits of holonomy representations, and if so, what constraints do they impose on the underlying differentials?
• RQ5 What is the rate of divergence of translation lengths in the holonomy representation as the norm of the quadratic differential increases?

Key findings

• For any divergent sequence of quadratic differentials with projective limit φ, any accumulation point of the holonomy map in the Morgan-Shalen boundary is represented by an R-tree admitting an equivariant, surjective, straight map from the R-tree Tφ dual to φ.
• If the projective limit φ has only simple zeros, then the holonomy sequence converges to the length function associated with the dual R-tree Tφ, implying uniqueness of the limiting action up to equivariant isometry.
• In the case of abelian length functions ℓ(γ) = |χ(γ)|, the underlying differential must be the square of a holomorphic 1-form ω whose imaginary part is the harmonic representative of the cohomology class [χ] ∈ H¹(X, ℝ).
• The translation length of any group element in the holonomy representation hol(φ) acting on H³ is O(||φ||^{1/2}), and this bound is sharp: some element achieves at least c||φ||^{1/2} for a uniform c > 0.
• The holonomy map Q(X) → X(Π) is properly embedded with an effective growth estimate, as the scale of translation lengths is comparable to ||φ||^{1/2}.
• The existence of sequences converging to abelian length functions with fixed Riemann surface remains an open question, though such sequences exist in higher genus with varying surfaces.