Teichmuller Space Research Articles

CLUSTER STRUCTURES ON HIGHER TEICHMULLER SPACES FOR CLASSICAL GROUPS

Let $S$ be a surface, $G$ a simply connected classical group, and $G^{\prime }$ the associated adjoint form of the group. We show that the moduli spaces of framed local systems ${\mathcal{X}}_{G^{\prime },S}$ and ${\mathcal{A}}_{G,S}$, which were constructed by Fock and Goncharov [‘Moduli spaces of local systems and higher Teichmuller theory’, Publ. Math. Inst. Hautes Études Sci.103 (2006), 1–212], have the structure of cluster varieties, and thus together form a cluster ensemble. This simplifies some of the proofs in that paper, and also allows one to quantize higher Teichmuller space, which was previously only possible when $G$ was of type $A$.

Cusp shape and tunnel number

We show that the set of cusp shapes of hyperbolic tunnel number one manifolds is dense in the Teichmuller space of the torus. A similar result holds for tunnel number n manifolds. As a consequence, for fixed n, there are infinitely many hyperbolic tunnel number n manifolds with at most one exceptional Dehn filling. This is in contrast to large volume Berge knots, which are tunnel number one manifolds, but with cusp shapes converging to a single point in Teichmuller space.

Non-convex balls in the Teichmüller metric

We prove that the Teichmuller space of surfaces of genus $g$ with $p$ punctures contains balls which are not convex in the Teichmuller metric whenever its complex dimension $(3g −3+p)$ is greater than $1$.

Symmetric properties of p-integrable Teichmüller spaces

We prove symmetric properties of the p-integrable subspaces \(T_p\) of the universal Teichmuller space T, for \(p>0.\) The elements of \(T_p\) are quasisymmetric automorphisms of the real line or of the unit circle, which have a q.c. extension with p-integrable complex dilatation with respect to the Poincare metric. We show that \(T_p\subset T_S,\) where \(T_S\) is the symmetric Teichmuller space and that, in the case of the real line, \(T_p\subset T_0,\) where \(T_0\) is the little Teichmuller space. The proofs use properties of the boundary behavior of q.c maps such as uniform weak conformality.

Besov Functions and Tangent Space to the Integrable Teichmüller Space

The authors identify the function space which is the tangent space to the integrable Teichmuller space. By means of quasiconformal deformation and an operator induced by a Zygmund function, several characterizations of this function space are obtained.

Uniformizations of stable $$(\gamma ,n)$$ ( γ , n ) -gonal Riemann surfaces

A $$(\gamma ,n)$$ -gonal pair is a pair (S, f), where S is a closed Riemann surface and $$f:S \rightarrow R$$ is a degree n holomorphic map onto a closed Riemann surface R of genus $$\gamma $$ . If the signature of (S, f) is of hyperbolic type, then it admits a uniformizing pair $$(\varGamma ,G)$$ , where G is a Fuchsian group acting on the unit disc $${{\mathbb {D}}}$$ containing $$\varGamma $$ as an index n subgroup, such that f is induced by the inclusion $$\varGamma \le G$$ . The uniformizing pair is uniquely determined by (S, f), up to conjugation by holomorphic automorphisms of $${{\mathbb {D}}}$$ , and it permits to provide a natural complex orbifold structure on the Hurwitz space parametrizing (twisted) isomorphic classes of pairs topologically equivalent to (S, f). In order to produce certain compactifications of these Hurwitz spaces, one needs to consider the so called stable $$(\gamma ,n)$$ -gonal pairs, which are natural geometrical deformations of $$(\gamma ,n)$$ -gonal pairs. Due to the above, it seems interesting to search for uniformizations of stable $$(\gamma ,n)$$ -gonal pairs, in terms of certain class of Kleinian groups. In this paper we review such uniformizations by using noded Fuchsian groups, obtained from the noded Beltrami differentials of Fuchsian groups that were previously studied by Alexander Vasil’ev and the author, and which provide uniformizations of stable Riemann orbifolds. These uniformizations permit to obtain a compactification of the Hurwitz spaces together a complex orbifold structure, these being quotients of the augmented Teichmuller space of G by a suitable finite index subgroup of its modular group.

Holomorphic Contractibility of Teichmüller Spaces

The problem of holomorphic contractibilty of Teichmuller spaces $${\mathbf {T}}(0, n)$$ of the punctures spheres ( $$n > 4$$ ) arose in the 1970s in connection with solving the algebraic equations in Banach algebras. We provide a positive solution of this problem.

Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths

For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations, and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from our previous paper, removing unnecessary assumptions on the surface.

On Kawai theorem for orbifold Riemann surfaces

We prove a generalization of Kawai theorem for the case of orbifold Riemann surface. The computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the Teichmuller space to the $${\mathrm {PSL}}(2,\mathbb {C})$$ -character variety, which allows to evaluate explicitly the pullback of Goldman symplectic form in the spirit of Riemann bilinear relations. As a corollary, we obtain a generalization of Goldman’s theorem that the pullback of Goldman symplectic form on the $${\mathrm {PSL}}(2,\mathbb {R})$$ -character variety is a symplectic form of the Weil–Petersson metric on the Teichmuller space.

On the postcritical set of a rational map

The postcritical set P(f) of a rational map $$f:{\mathbb P}^1\rightarrow {\mathbb P}^1$$ is the smallest forward invariant subset of $${\mathbb P}^1$$ that contains the critical values of f. In this paper we show that every finite set $$X\subset {\mathbb P}^1({\overline{{\mathbb Q}}})$$ can be realized as the postcritical set of a rational map. We also show that every map $$F:X\rightarrow X$$ defined on a finite set $$X\subset {\mathbb P}^1({\mathbb C})$$ can be realized by a rational map $$f:P(f)\rightarrow P(f)$$, provided we allow small perturbations of the set X. The proofs involve Belyi’s theorem and iteration on Teichmuller space.

On [formula omitted]-Teichmüller spaces

Recently, Wulan and Ye [36] introduced the QK-Teichmüller spaces and proved some theorems about the theory of QK-Teichmüller spaces. In this note, we continue to consider this theory and obtain some results on the QK-Teichmüller spaces. In particular, we establish some new characterizations of the little QK-Teichmüller spaces.

Length spectrum characterization of asymptotic Teichmüller space

Let $$S_0$$ be a Riemann surface of infinite type without boundary and admitting an upper and lower bounded pair of pants decomposition, and let $${ AT}(S_0)$$ be the asymptotic Teichmuller space of S. In this paper, we first give a length spectrum definition of such a Teichmuller space $${ AT}(S_0)$$ . Then we introduce a length spectrum metric on $${ AT}(S_0)$$ and show that it induces the same topology on $${ AT}(S_0)$$ as the Teichmuller metric. We also show that these two metrics are not bi-Lipschitz.

Hamiltonian properties of earthquake flows on surfaces with closed geodesic boundary

The Teichmuller space $${\mathcal {T}}_S(\mathbf {b})$$ of hyperbolic metrics on a surface S with fixed lengths at the boundary components is symplectic. We prove that any sum of infinitesimal earthquakes on S that is tangent to $${\mathcal {T}}_S(\mathbf {b})$$ is Hamiltonian, by providing a Hamiltonian $$\mathbb {L}$$ . Such function extends the classical length map associated to a compactly supported measured geodesic lamination and shares with it some peculiar properties, such as properness and strict convexity along earthquakes paths under usual topological conditions. As an application, we prove that any non-Fuchsian affine representation of $$\pi _1(S)$$ into $$\mathbb {R}^{2,1}\rtimes SO_0(2,1)$$ with cocompact discrete linear part is determined by the singularities of the two invariant regular domains in $$\mathbb {R}^{2,1}$$ pointed out by Barbot, once the boundary lengths are fixed.

Determinant Bundle Over the Universal Moduli Space of Principal Bundles Over the Teichmüller Space

For a Riemann surface X and the moduli of regularly stable G-bundles M, there is a naturally occuring “adjoint” vector bundle over X × M. One can take the determinant of this vector bundle with respect to the projection map onto M. Our aim here is to study the curvature of the determinant bundle as the conformal structure on X varies over the Teichmuller space.

Carathéodory’s metrics on Teichmüller spaces and L-shaped pillowcases

One of the most important results in Teichmuller theory is Royden’s theorem, which says that the Teichmuller and Kobayashi metrics agree on the Teichmuller space of a given closed Riemann surface. The problem that remained open is whether the Caratheodory metric agrees with the Teichmuller metric as well. In this article, we prove that these two metrics disagree on each T_g, the Teichmuller space of a closed surface of genus g ≥ 2. The main step is to establish a criterion to decide when the Teichmuller and Caratheodory metrics agree on the Teichmuller disk corresponding to a rational Jenkins–Strebel differential φ. First, we construct a holomorphic embedding ℰ:H^k → Tg,n corresponding to φ. The criterion says that the two metrics agree on this disk if and only if a certain function Φ: ℰ (H^k) → H can be extended to a holomorphic function Φ : T_(g,n) → H. We then show by explicit computation that this is not the case for quadratic differentials arising from L-shaped pillowcases.

Koebe Spaces of Infinite Type

Let D be a Koebe circle domain in the complex plane of infinite type. We show that, when D is tame, D corresponds to an infinitely generated extended classical Schottky group G and the quasiconformal Koebe space of D can be identified with the quasiconformal deformation space of G. Moreover, when D also satisfies the boundedness conditions, we prove that the essential Teichmuller space of the G-orbit of \(\infty \) admits the complex analytic structure.

Toy Teichmüller spaces of real dimension 2: the pentagon and the punctured triangle

We study two 2-dimensional Teichmuller spaces of surfaces with boundary and marked points, namely, the pentagon and the punctured triangle. We show that their geometry is quite different from Teichmuller spaces of closed surfaces. Indeed, both spaces are exhausted by regular convex geodesic polygons with a fixed number of sides, and their geodesics diverge at most linearly.

Almost isometries between Teichmüller spaces

Almost isometries between Teichmüller spaces

The arc metric on Teichmüller spaces of surfaces of infinite type with boundary

Let $X_{0}$ be a complete hyperbolic surface of infinite type with geodesic boundary which admits a countable pair of pants decomposition. As an application of the Basmajian identity for complete bordered hyperbolic surfaces of infinite type with limit sets of 1-dimensional measure zero, we define an asymmetric metric (which is called arc metric) on the quasiconformal Teichmuller space $\mathcal{T}(X_{0})$ provided that $X_{0}$ satisfies a geometric condition. Furthermore, we construct several examples of hyperbolic surfaces of infinite type satisfying the geometric condition and discuss the relation between the Shiga's condition and the geometric condition.

On the completeness of the asymptotic length spectrum Teichmüller space of surfaces of infinite type

The length spectrum Teichmuller space \(T_{ls}(R)\), based at hyperbolic surface of infinite type R, is the collection of all marked hyperbolic surfaces that are homeomorphic to R, satisfying that the quotients of the lengths of the corresponding geodesics are uniformly bounded from above and from below. Two points in \(T_{ls}(R)\) are called asymptotically length spectrum equivalent if the ratios of the lengths of geodesics outside compact sets is close to 1. The quotient space coming from this relation is called the asymptotic length spectrum Teichmuller space \(AT_{ls}(R)\). In this paper we prove that if the base surface R admits a pair of pants decomposition that satisfies Shiga’s condition (i.e., that is upper and lower bounded), then \(AT_{ls}(R)\) is complete under the natural metric. We also prove that in this case, the space is homeomorphic to \(l^{\infty }/c_0\), where \(l^{\infty }\) is the Banach space of bounded sequences and \(c_0\) is the subspace of sequences converging to zero.