[Paper Review] A Hyperelliptic View on Teichmuller Space. I

Sasha Anan’in, Eduardo C. Bento Gonçalves|ArXiv.org|Sep 11, 2007

Geometric and Algebraic Topology9 references3 citations

TL;DR

This paper presents an elementary, geometric approach to Teichmüller space using hyperelliptic Riemann surfaces and simple earthquakes—continuous deformations along geodesics—proving that the earthquake group acts transitively on the Teichmüller space of hyperelliptic surfaces. It establishes a simple, effective criterion for faithfulness and discreteness of representations via natural boundary coordinates, and extends the framework to general Riemann surfaces, showing that Teichmüller space fibers over the hyperelliptic locus with unique projections.

ABSTRACT

We explicitly describe the Teichmuller space TH_n of hyperelliptic surfaces in terms of natural and effective coordinates as the space of certain (2n-6)-tuples of distinct points on the ideal boundary of the Poincare disc. We essentially use the concept of a simple earthquake which is a particular case of a Fenchel-Nielsen twist deformation. Such earthquakes generate a group that acts transitively on TH_n. This fact can be interpreted as a continuous analog of the well-known Dehn theorem saying that the mapping class group is generated by Dehn twists. We find a simple and effective criterion that verifies if a given representation of the surface group π_1Σin the group of isometries of the hyperbolic plane is faithful and discrete. The article also contains simple and elementary proofs of several known results, for instance, of W. M. Goldman's theorem [Gol1] characterizing the faithful discrete representations as having maximal Toledo invariant (which is essentially the area of the representation in the two-dimensional case).

Motivation & Objective

• To develop an elementary, geometric framework for studying Teichmüller spaces without relying on analytic methods.
• To characterize the Teichmüller space of hyperelliptic Riemann surfaces using natural boundary coordinates on the ideal boundary of the Poincaré disk.
• To prove that the earthquake group acts transitively on the Teichmüller space of hyperelliptic surfaces, providing a continuous analog of Dehn's theorem.
• To extend the method to general Riemann surfaces and establish a criterion for faithfulness and discreteness of representations via a natural fundamental domain.
• To lay the foundation for a complex hyperbolic analog of the theory, particularly for Toledo's theorem and discrete group representations.

Proposed method

• Represent hyperelliptic surfaces as configurations of $ n = 2g+2 $ points in the Poincaré disk, with reflections generating the fundamental group $ H_n $.
• Define simple earthquakes (SEs) as continuous deformations preserving distance between adjacent points along a geodesic, generating the earthquake group $ ilde{ m E}_n $.
• Use the cyclic order of $ 2n-6 $ distinct boundary points to parametrize the Teichmüller space $ ilde{ m T}H_n $, providing natural, geometric coordinates.
• Apply Poincaré’s Polyhedron Theorem to show that a polygon $ Q $ with $ 2(n-2) $ vertices and area $ 2(n-4)π $ is a fundamental domain for the group action.
• Construct a fundamental domain for general $ G_n = π_1(Σ) $, allowing visualization of the universal family of Riemann surfaces over $ ilde{ m T}_n $.
• Extend the earthquake action from $ ilde{ m T}H_n $ to $ ilde{ m T}_n $ by defining $ ho E_i(t) $ as a conjugation twist along hyperbolic elements in the representation.

Experimental results

Research questions

• RQ1 Can Teichmüller space for hyperelliptic surfaces be parametrized using natural, geometric coordinates on the boundary of the Poincaré disk?
• RQ2 Does the earthquake group $ ilde{ m E}_n $ act transitively on the Teichmüller space $ ilde{ m T}H_n $, and can this be seen as a continuous analog of Dehn's theorem?
• RQ3 Is there a simple, effective criterion to determine whether a representation $ \varrho: H_n \to \mathrm{PU}(1,1) $ is faithful and discrete?
• RQ4 Can the methods used for hyperelliptic surfaces be extended to general Riemann surfaces to yield a criterion for discreteness of representations?
• RQ5 How does the structure of $ \tilde{ m T}_n $ relate to $ \tilde{ m T}H_n $, and can $ \tilde{ m T}_n $ be fibered over $ \tilde{ m T}H_n $?

Key findings

• The Teichmüller space $ \tilde{ m T}H_n $ of a hyperelliptic surface of genus $ g \geq 2 $ is parametrized by $ 2n-6 $ distinct points on the ideal boundary $ \partial\mathbb{D} $, ordered cyclically.
• The earthquake group $ \tilde{ m E}_n $ acts transitively on $ \tilde{ m T}H_n $, providing a continuous analog of the Dehn twist generation of the mapping class group.
• For $ n=5 $, every nontrivial representation $ \varrho: H_5 \to \mathrm{PU}(1,1) $ is automatically faithful and discrete, a result not previously noted in the literature.
• A representation $ \varrho: G_n \to \mathrm{PU}(1,1) $ is discrete and faithful if and only if the associated fundamental polygon $ Q $ has area $ 2(n-4)\pi $, with interior angles summing to $ 2\pi $.
• The universal family of Riemann surfaces $ \mathcal{F} \to \tilde{ m T}_n $ is visualized via a fundamental domain in $ \mathbb{D} \times \tilde{ m T}_n $, with fibers isomorphic to $ \mathbb{D}/G_n $.
• Teichmüller space $ \tilde{ m T}_n $ is fibered twice over $ \tilde{ m T}H_n $, and every point in $ \tilde{ m T}_n $ is uniquely determined by its projections to $ \tilde{ m T}H_n $.