Teichmuller Space Research Articles

Quakebend deformations in complex hyperbolic quasi-Fuchsian space

We study quakebend deformations in complex hyperbolic quasi-Fuchsian space QC.U/ of a closed surface U of genus g> 1, that is the space of discrete, faithful, totally loxodromic and geometrically finite representations of the fundamental group of U into the group of isometries of complex hyperbolic space. Emanating from an R‐Fuchsian point 2QC.U/, we construct curves associated to complex hyperbolic quakebending of and we prove that we may always find an open neighborhood U./ of in QC.U/ containing pieces of such curves. Moreover, we present generalisations of the well known Wolpert‐Kerckhoff formulae for the derivatives of geodesic length function in Teichmuller space. 32G05; 32M05

Projective structures, grafting and measured laminations

We show that grafting any fixed hyperbolic surface defines a homeomorphism from the space of measured laminations to Teichmuller space, complementing a result of Scannell-Wolf on grafting by a fixed lamination. This result is used to study the relationship between the complex-analytic and geometric coordinate systems for the space of complex projective ($\CP^1$) structures on a surface. We also study the rays in Teichmuller space associated to the grafting coordinates, obtaining estimates for extremal and hyperbolic length functions and their derivatives along these grafting rays.

Tropicalization of group representations

In this paper we give an interpretation to the boundary points of the compactification of the parameter space of convex projective structures on an n-manifold M. These spaces are closed semi-algebraic subsets of the variety of characters of representations of the fundamental group of M in SL_{n+1}(R). The boundary was constructed as the tropicalization of this semi-algebraic set. Here we show that the geometric interpretation for the points of the boundary can be constructed searching for a tropical analogue to an action of the group on a projective space. To do this we need to construct a tropical projective space with many invertible projective maps. We achieve this using a generalization of the Bruhat-Tits buildings for SL_{n+1} to non-archimedean fields with real surjective valuation. In the case n = 1 these objects are the real trees used by Morgan and Shalen to describe the boundary points for the Teichmuller spaces. In the general case they are contractible metric spaces with a structure of tropical projective spaces.

An extension of the Weil–Petersson metric to quasi-Fuchsian space

We define a natural semi-definite metric on quasi-fuchsian space, derived from geodesic current length functions and Hausdorff dimension, that extends the Weil–Petersson metric on Teichmuller space. We use this to describe a metric on Teichmuller space obtained by taking the second derivative of Hausdorff dimension and show that this metric is bounded below by the Weil–Petersson metric. We relate the change in Hausdorff dimension under bending along a measured lamination to the length in the Weil–Petersson metric of the associated earthquake vector of the lamination.

Teichmüller theory of the punctured solenoid

The punctured solenoid $${\mathcal H}$$ plays the role of an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmuller space of $${\mathcal H}$$ is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of $${\mathcal H}$$ . Furthermore, a point in the decorated Teichmuller space induces a polygonal decomposition of $${\mathcal H}$$ itself giving a combinatorial description of the decorated Teichmuller space. This is used to obtain a non-trivial set of generators of the modular group of $${\mathcal H}$$ , and each word in these generators admits a normal form. There is furthermore a non-degenerate modular group invariant two form on the Teichmuller space of $${\mathcal H}$$ . All of this structure is in perfect analogy with that of the decorated Teichmuller space of a punctured surface of finite type.

A Survey of Total Positivity

This paper contains an exposition of the classical work of Schoenberg and Gantmacher, Krein on the theory of totally positive matrices. It also contains a generalization of this theory to the Lie theoretic context and an application to the recent results of Fock and Goncharov on Teichmuller theory.

Teichmüller geodesics that do not have a limit in 𝒫ℳℱ

We construct a Teichmuller geodesic which does not have a limit on the Thurston boundary of the Teichmuller space.

Dynamics on Teichmüller spaces and self-covering of Riemann surfaces

A non-injective holomorphic self-cover of a Riemann surface induces a non-surjective holomorphic self-embedding of its Teichmuller space. We investigate the dynamics of such self-embeddings by applying our structure theorem of self-covering of Riemann surfaces and examine the distribution of its isometric vectors on the tangent bundle over the Teichmuller space. We also extend our observation to quasiregular self-covers of Riemann surfaces and give an answer to a certain problem on quasiconformal equivalence to a holomorphic self-cover.

On quasiconformal deformations of the universal hyperbolic solenoid

We investigate the Teichmuller metric and the complex structure on the Teichmuller space \( \mathcal{T} \)(H∞) of the universal hyperbolic solenoid H∞. In particular, a version of the Reich-Strebel inequality for H∞ is obtained. As a consequence, we show that the Teichmuller type Beltrami coefficients determine unique geodesics in \( \mathcal{T} \)(H∞), and we compute the infinitesimal form of the Teichmuller metric. In addition, we show that a Beltrami coefficient is Teichmuller extremal if and only if it is infinitesimally extremal. Finally, we show that the Kobayashi metric on \( \mathcal{T} \)(H∞) equals the Teichmuller metric.

Comparison between Teichmüller and Lipschitz metrics

AbstractWe study the Lipschitz metric on a Teichmuller space (defined by Thurston) and compare it with the Teichmuller metric. We show that in the thin part of the Teichmuller space the Lipschitz metric is approximated up to a bounded additive distortion by the sup-metric on a product of lower-dimensional spaces (similar to the Teichmuller metric as shown by Minsky). In the thick part, we show that the two metrics are equal up to a bounded additive error. However, these metrics are not comparable in general; we construct a sequence of pairs of points in the Teichmuller space, with distances that approach zero in the Lipschitz metric while they approach infinity in the Teichmuller metric.

Quantum Teichmüller spaces and Kashaev’s 6j–symbols

The Kashaev invariants of 3‐manifolds are based on 6j ‐symbols from the representation theory of the Weyl algebra, a Hopf algebra corresponding to the Borel subalgebra of Uq.sl.2;C//. In this paper, we show that Kashaev’s 6j ‐symbols are intertwining operators of local representations of quantum Teichmuller spaces. This relates Kashaev’s work with the theory of quantum Teichmuller space, which was developed by Chekhov‐Fock, Kashaev and continued by Bonahon‐Liu. 57R56; 20G42

Earthquakes and Thurston’s boundary for the Teichmüller space of the universal hyperbolic solenoid

Earthquakes and Thurston’s boundary for the Teichmüller space of the universal hyperbolic solenoid

A uniqueness property for the quantization of Teichmüller Spaces

Chekhov, Fock and Kashaev introduced a quantization of the Teichmuller space of a punctured surface S, and an exponential version \({\mathcal{T}}^q(S)\) of this construction was developed by Bonahon and Liu. The construction of \({\mathcal{T}}^q(S)\) crucially depends on certain coordinate change isomorphisms between the Chekhov–Fock algebras associated to different ideal triangulations of S. We show that these coordinate change isomorphisms are essentially unique, once we require them to satisfy a certain number of natural conditions.

Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces

For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichmuller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichmuller space AT(R). We prove that if MCG(R) has a common fixed point α(p) ∈ AT(R), then it acts discontinuously on the fiber T p over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T o = T(R) for an analytically finite Riemann surface R.

Some Properties of an Operator on L ∞(Δ) and Its Applications

In this paper, we introduce an operator H μ (z) on L ∞(Δ) and obtain some of its properties. Some applications of this operator to the extremal problem of quasiconformal mappings are given. In particular, a sufficient condition for a point τ in the universal Teichmuller space T(Δ) to be a Strebel point is obtained.

On the variation of a series on Teichmüller space

On the variation of a series on Teichmüller space

Extensions of holomorphic motions

We prove that a normalized holomorphic motion of a closed set E is induced by a holomorphic map into the Teichmuller space of E, denoted by T(E), if and only if it can be extended to a normalized continuous motion of the Riemann sphere. We also prove that the extension can be chosen to have additional properties.

Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms

We investigate the representation theory of the polynomial core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional representations of this polynomial core are classified, up to finitely many choices, by group homomorphisms from the fundamental group of the surface to the isometry group of the hyperbolic 3--space. We exploit this connection between algebra and hyperbolic geometry to exhibit new invariants of diffeomorphisms of S.

Teichmüller Theory of Bordered Surfaces

We propose the graph description of Teichmuller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formu- late this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordi- nates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichmuller spaces of bordered surfaces of the same genus, same number of boun- dary components, and same total number of marked points but with arbitrary distributions of marked points among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras and discuss briefly the relation to the Thurston theory.

A quasiconformal mapping class group acting trivially on the asymptotic Teichmüller space

For an analytically infinite Riemann surface R, the quasiconformal mapping class group MCG(R) always acts faithfully on the ordinary Teichmuller space T(R). However in this paper, an example of R is constructed for which MCG(R) acts trivially on its asymptotic Teichmuller space AT(R).