We prove the identity $$ \prod_γ\left(\frac{e^{l(γ)}+1}{e^{l(γ)}-1}\right)^{2h}=\exp\left(\frac{l_1+l_2+l_3}{2}\right), $$ (or $$ \prod_γ\left(\frac{t(γ)^2}{t(γ)^2-4}\right)^h=\frac{t_1+\sqrt{t_1^2-4}}{2}\cdot\frac{t_2+\sqrt{t_2^2-4}}{2}\cdot\frac{t_3+\sqrt{t_3^2-4}}{2} $$ in trace coordinates), where the product is over all simple closed geodesics on the once-punctured torus, $l(γ)=2\operatorname{arccosh}(t(γ)/2)$ is the length of the geodesic, and $l_i$ ($t_i$) are the lengths (traces) of any triple of simple geodesics $\{γ_i\}$ intersecting at a single point. The exponent $h=h(γ;\{γ_i\})$ is a positive integer "height" which increases as we move away from the chosen triple $\{γ_i\}$ in its orbit under $SL_2(\mathbb{Z})$ (see Figure 1 for the "definition by picture"). For comparison, a short proof of McShane's identity $$ \sum_γ\frac{1}{1+e^{l(γ)}}=\frac{1}{2}=\sum_γ\frac{1-\sqrt{1-4/t(γ)^2}}{2} $$ in the same spirit is given in an appendix. Both proofs are elementary and proceed by "integrating" around the chosen triple $\{γ_i\}$ in its Teichmüller orbit.
In this paper, we introduce a new asymmetric weak metric on the Teichm{ü}ller space of a closed orientable surface with (possibly empty) punctures.This new metric, which we call the Teichm{ü}ller-Randers metric, is an asymmetric deformation of the Teichm{ü}ller metric, and is obtained by adding to the infinitesimal form of the Teichm{ü}ller metric a differential 1-form. We study basic properties of the Teichm{ü}ller-Randers metric. In the case when the 1-form is exact, any Teichm{ü}ller geodesic between two points is a unique Teichm{ü}ller--Randers geodesic between them. A particularly interes…
We comment on Teichm{ü}ller 's paper ''Vollst{ä}ndige L{ö}sung einer Extremalaufgabe der quasikonformen Abbildung'' (Complete solution of an ex-tremal problem of the quasiconformal mapping),, published in 1941. In this paper, Teichm{ü}ller gives a proof of the existence of extremal quasiconformal mappings in the case of the pentagon (disc with five distinguished points on the boundary). The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichm{ü}ller theory.
In this paper, the Teichm{ü}ller spaces of surfaces appear from two points of views: the conformal category and the hyperbolic category. In contrast to the case of surfaces of topologically finite type, the Teichm{ü}ller spaces associated to surfaces of topologically infinite type depend on the choice of a base structure. In the setting of surfaces of infinite type, the Teichm{ü}ller spaces can be endowed with different distance functions such as the length-spectrum distance, the bi-Lipschitz distance, the Fenchel-Nielsen distance, the Teichm{ü}ller distance and there are other distance functi…
For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest Lipschitz constant in the homotopy class of this map relative to the boundary. As a consequence of this construction, we exhibit new geodesics for the arc metric on the Teichm{ü}ller space of an arbitrary surface of negative Euler characteristic with nonempty boundary. We also obtain new geodesics for Thurston's metric on Teichm{ü}ller spaces of hyperbolic surfaces without boundary.
These are notes on the hyperbolic geometry of surfaces, Teichm{ü}ller spaces and Thurston's metric on these spaces. They are associated with lectures I gave at the Morningside Center of Mathematics of the Chinese Academy of Sciences in March 2019 and at the Chebyshev Laboratory of the Saint Petersburg State University in May 2019. In particular, I survey several results on the behavior of stretch lines, a distinguished class of geodesics for Thurston's metric and I point out several analogies between this metric and Teichm{ü}ller's metric. Several open questions are addressed. The final versio…
In this paper, we show that the analogue of Thurston's asymmetric metric on the Teichm{ü}ller space of flat structures on the torus is weak Finsler and we give a geometric description of its unit sphere at each point in the tangent space to Teichm{ü}ller space. We then introduce a family of weak Finsler metrics which interpolate between Thurston's asymmetric metric and the Teichm{ü}ller metric of the torus (which coincides with the the hyperbolic metric). We describe the infinitesimal unit spheres of the metrics in this family.The final version of this paper will appear in Annales Academiæ \ S…
Let $S$ be a closed oriented surface of genus at least $2$, and denote by $\mathcal{T}(S)$ its Teichm{ü}ller space. For any isotopy class of closed curves $γ$, we compute the first three derivatives of the length function $\ell\_γ:\mathcal{T}(S)\rightarrow\mathbf{R}\_+$ in the shearing coordinates associated to a maximal geodesic lamination $λ$. We show that if $γ$ intersects every leaf of $λ$, then the Hessian of $\ell\_γ$ is positive-definite. We extend this result to length functions of measured laminations. We also provide a method to compute higher derivatives of length functions of geode…
We survey some major contributions to Riemann's moduli space and Teichm{ü}ller space. Our report has a historical character, but the stress is on the chain of mathematical ideas. We start with the introduction of Riemann surfaces, and we end with the discovery of some of the basic structures of Riemann's moduli space and Teichm{ü}ller space. We point out several facts which seem to be unknown to many algebraic geometers and analysts working in the theory. The period we are interested in starts with Riemann, in 1851, and ends in the early 1960s, when Ahlfors and Bers confirmed that Teichm{ü}lle…
This is a commentary on Teichm{ü}ller's paper Unter-suchungen{ü}ber konforme und quasikonforme Abbildungen (Inves-tigations on conformal and quasiconformal mappings) published in 1938. The paper contains fundamental results in conformal geometry , in particular a lemma, known as the Modulsatz, which insures the almost circularity of certain loci defined as complementary components of simply connected regions in the Riemann sphere, and another lemma, which we call the Main Lemma, which insures the circularity near infinity of the image of circles by a qua-siconformal map. The two results find w…
The type problem is the problem of deciding, for a simply connected Riemann surface, whether it is conformally equivalent to the complex plane or to the unit dic in the complex plane. We report on Teichm{ü}ller's results on the type problem from his two papers Eine Anwendung quasikonformer Abbildungen auf das Typenproblem (An application of quasiconformal map-pings to the type problem) (1937) and Untersuchungen{ü}ber kon-forme und quasikonforme Abbildungen (Investigations on con-formal and quasiconformal mappings) (1938). They concern simply connected Riemann surfaces defined as branched cover…
This is a mathematical commentary on Teichm{ü}ller's paper ``Bestimmung der extremalen quasikonformen Abbildungen bei geschlossenen orientierten Riemannschen Fl{ä}chen'' (Determination of extremal quasiconformal maps of closed oriented Riemann surfaces). This paper is among the last (and may be the last one) that Teichm{ü}ller wrote on the theory of moduli. It contains the proof of the so-called Teichm{ü}ller existence theorem for a closed surface of genus at least 2. For this proof, the author defines a mapping between a space of equivalence classes of marked Riemann surfaces (the Teichm{ü}ll…
We provide a commentary on Teichm{ü}ller's paper "Extremale quasikonforme Abbildungen und quadratische Differentiale" (Extremal quasiconformal mappings of closed oriented Riemann surfaces), Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl. 1940, No.22, 1-197 (1940). The paper is quoted in several works, although it was read by very few people. Some of the results it contains were rediscovered later on and published without any reference to Teichm{ü}ller.In this commentary, we highlight the main results and the main ideas contained in that paper and we describe some of the important developments they…
The combinatorial structure of the unit tangent spheres and cotangent spheres of Teichm{ü}ller space with Thurston's Finsler metric
Gakushuin University（学习院大学）；Max-Planck Institute for Mathematics（马克斯·普朗克数学研究所）；Institut de Recherche Mathématique Avancée, CNRS et Université de Strasbourg（斯特拉斯堡大学法国国家科学研究中心高级数学研究所）
For $g \ge 5$, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building off of work of the first author (arXiv:1901.05482), we find that the non-hyperelliptic components are classified by an invariant known as an $r$-spin structure. This is accomplished by computing a certain monodromy group valued in the mapping class group. To do this, we determine explicit finite generating sets for all $r$-spin stabilizer subgroup…
This is a commentary on Teichm{ü}ller's paper Ein Verschiebungssatz der quasikonformen Abbildung (A displacement theorem of quasiconformal mapping), published in 1944. We explain in detail how Teichm{ü}ller solves the problem of finding the quasiconformal mapping from the unit disc to itself, sending 0 to a strictly negative point on the real line, holding the boundary of the disc pointwise fixed and with the smallest quasiconformal dilatation. We mention also some consequences of this extremal problem and we ask a question.
Wolf gave a homeomorphism from the Teichmüller space to the space of quadratic differentials on a closed Riemann surface by using harmonic maps. Moreover, using harmonic maps rays, he gave a compactification of the Teichmüller space and show that it coincides with the Thurston compactification. In this paper, we extend the harmonic maps compactification to the Teichmüller spaces of punctured Riemann surfaces, and show that it still coincides with the Thurston compactification.
The moduli space of lattices of $\mathbb{C}$ is a Riemann surface of finite hyperbolic area with the square lattice as an origin. We select a lattice from the induced uniform distribution and calculate the statistics of the Teichmüller distance to the origin. This in turn identifies distribution of the distance in Teichmüller space to the central "square" punctured torus in the moduli space of punctured tori. There are singularities in this p.d.f. arising from the topology of the moduli space. We also consider the statistics of the distance in Teichmüller space to the rectangular punctured tor…
The arc metric is an asymmetric metric on the Teichm{ü}ller space T(S) of a surface S with nonempty boundary. In this paper we study the relation between Thurston's compactification and the horofunction compactification of T(S) endowed with the arc metric. We prove that there is a natural homeomorphism between the two compactifications.
The punctured solenoid $§$ is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller space of $§$ is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of $§$. Furthermore, a point in the decorated Teichmüller space induces a polygonal decomposition of $§$ giving a combinatorial description of its decorated Teichmüller space itself. This is used to obtain a non-trivial set of generators of the modular group of $§$, which is presumably the main res…
We study two $2$-dimensional Teichmüller spaces of surfaces with boundary and marked points, namely, the pentagon and the punctured triangle. We show that their geometry is quite different from Teichmüller 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.
We construct new coordinates for the Teichmüller space Teich of a punctured torus into $\bold{R} \times\bold{R}^+$. The coordinates depend on the representation of Teich as a space of marked Kleinian groups $G_μ$ that depend holomorphically on a parameter $μ$ varying in a simply connected domain in $\bold{C}$. They describe the geometry of the hyperbolic manifold $\bold{H}^3/G_μ$; they reflect exactly the visual patterns one sees in the limit sets of the groups $G_μ$; and they are directly computable from the generators of $G_μ$.
We consider the following question: Which parameters in the extension of a rational pleating ray across the boundary of $\Cal M$, the Maskit embedding of the Teichmüller space of once punctured tori correspond to a Kleinian group? Using methods of Keen and Series and Wright we prove a local result, stating that on each rational ray there is a sequence of parameters in $\overline\Bbb H\setminus\Cal M$ accumulating at the boundary point of $\Cal M$ on the ray. These are the unique parameters on the extended $p/q$ ray for which the special word $W_{p/q}$ is a primitive elliptic Möbius transformat…
Hamenstädt gave a parametrization of the Teichmüller space of punctured surfaces such that the image under this parametrization is the interior of a polytope. In this paper, we study the Hilbert metric on the Teichmüller space of punctured surfaces based on this parametrization. We prove that every earthquake ray is an almost geodesic under the Hilbert metric.
We consider complex Fenchel-Nielsen coordinates on the quasi-Fuchsian space of punctured tori. These coordinates arise from a generalisation of Kra's plumbing construction and are related to earthquakes on Teichmueller space. They also allow us to interpolate between two coordinate systems on Teichmueller space, namely the classical Fuchsian space with Fenchel-Nielsen coordinates and the Maskit embedding. We also show how they relate to the pleating coordinates of Keen and Series.
Previous work of the author has developed coordinates on bundles over the classical Teichmueller spaces of punctured surfaces and on the space of cosets of the Moebius group in the group of orientation-preserving homeomorphisms of the circle, and this work is surveyed here. Joint work with Dragomir Saric is also sketched which extends these results to the setting of the Teichmueller space of the solenoid of a punctured surface, which is defined here in analogy to Dennis Sullivan's original definition for the case of closed surfaces. Because of relations with the classical modular group, the pu…
We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in linear recurrence relations, the second in hyperbolic geometry. The two methods align, providing both an algebraic and geometric interpretation of the earthquake deformations. We convert the expressions to other coordinate systems for Teichmüller space to examine earthquake deformations further. Two families of curves are used as examples. E…
Grothendieck's Esquisse d'un programme is often referred to for the ideas it contains on dessins d'enfants, the Teichm{ü}ller tower, and the actions of the absolute Galois group on these objects or their etale fundamental groups. But this program contains several other important ideas. In particular, motivated by surface topology and moduli spaces of Riemann surfaces, Grothendieck calls there for a recasting of topology, in order to make it fit to the objects of semialgebraic and semianalytic geometry, and in particular to the study of the Mumford-Deligne compactifications of moduli spaces. A …
The Dynnikov coordinate system puts global coordinates on the boundary of Teichmüller space of an $n$--punctured disk. We survey the Dynnikov coordinate system, and investigate how we use this coordinate system to study pseudo--Anosov braids making use of results from Thurston's theory on surface homeomorphisms.
We give an example of a horocycle in the Teichmüller space of the five-times-punctured sphere that does not converge in the Gardiner--Masur compactification, or equivalently in the horofunction compactification of the Teichmüller metric. As an intermediate step, we exhibit a simple closed curve whose extremal length is periodic but not constant along the horocycle. The example lifts to any Teichmüller space of complex dimension greater than one via covering constructions.
We prove that the Teichmüller space of surfaces with given boundary lengths equipped with the arc metric (resp. the Teichmüller metric) is almost isometric to the Teichmüller space of punctured surfaces equipped with the Thurston metric (resp. the Teichmüller metric).
We show that the reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms of punctured surfaces are intertwiners of local representations of the quantum Teichmüller spaces. We characterize them as the only intertwiners that satisfy certain natural cut-and-paste operations of topological quantum field theories and such that their traces define invariants of mapping tori.
It was recently shown that the Carathéodory and Teichmüller metrics on the Teichmüller space of a closed surface do not coincide. On the other hand, Kra earlier showed that the metrics coincide when restricted to a Teichmüller disk generated by a differential with no odd-order zeros. Our aim is to classify Teichmüller disks on which the two metrics agree, and we conjecture that the Carathéodory and Teichmüller metrics agree on a Teichmüller disk if and only if the Teichmüller disk is generated by a differential with no odd-order zeros. Using dynamical results of Minsky, Smillie, and Weiss, we …
In 1986 William P. Thurston introduced the celebrated (asymmetric) Lipschitz distance on the Teichmueller space of a (closed or punctured) surface. In this paper we extend his work to the Teichmueller space of a surface with boundary endowed the arc distance. In this new setting we construct a large family of geodesics, which generalize Thurston's stretch lines. We prove that the Teichmueller space of a surface with boundary, endowed with the arc distance, is a geodesic metric space. Furthermore, the arc distance is induced by a Finsler metric. As a corollary, we describe a new class of geodes…
We give an irreducible decomposition of the so-called local representations (see arXiv:0707.2151) of the quantum Teichmüller space $\mathcal{T}_q(Σ)$ where $Σ$ is a punctured surface of genus $g>0$ and $q$ is a primitive $N$-th root of unity with $N$ odd. As an application, we construct a family of representations of the Kauffman bracket skein algebra of the closed surface $\overlineΣ$.
We consider the quantum Teichmuller space of the punctured surface introduced by Chekhov-Fock-Kashaev, and formalize it as a noncommutative deformation of the space of algebraic functions on the Teichmuller space of the surface. In order to apply it in 3-dimensional topology, we put more attention to the details involving small surfaces.
For a given $ε>0$, we show that there exist two finite index subgroups of $PSL_2(\mathbb{Z})$ which are $(1+ε)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $ε>0$ there are two finite regular covers of the Modular once punctured torus $T_0$ (or just the Modular torus) and a $(1+ε)$-quasiconformal between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space $T(§)$ of the punctured solenoid $§$ under the action of the corresponding Modu…
Chekhov, Fock and Kashaev introduced a quantization of the Teichmüller space $\mathcal{T}^q(S)$ of a punctured surface $S$, and an exponential version of this construction was developed by Bonahon and Liu. The construction of the quantum Teichmüller space 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.
We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has non-compact ends with boundary cusps. This extends Wolf's parametrization of the Teichmüller space of a closed surface using holomorphic quadratic differentials. Our proof involves showing the existence of a harmonic map from a punctured Riemann surface to a crowned hyperbolic surface, with prescribed principal parts of its Hopf different…
We construct a counterexample for an analogue of Masur's criterion in the setting of Teichmüller space equipped with the Thurston metric. For that, we find a minimal, filling, non-uniquely ergodic lamination $λ$ on the seven-times punctured sphere with uniformly bounded annular projection distances. Then we show that a geodesic in the corresponding Teichmüller space that converges to $λ$, stays in the thick part for the whole time.
This paper extends the decorated Teichmüller theory developed before for punctured surfaces to the setting of ``bordered'' surfaces, i.e., surfaces with boundary, and there is non-trivial new structure discovered. The main new result identifies the arc complex of a bordered surface up to proper homotopy equivalence with a certain quotient of the moduli space, namely, the quotient by the natural action of the positive reals by homothety on the hyperbolic lengths of geodesic boundary components. One tool in the proof is a homeomorphism between two versions of a ``decorated'' moduli space for bor…
We calculate the knot invariant coming from the Teichmüller TQFT [AK1]. Specifically we calculate the knot invariant for the complement of the knot $6_1$ both in the original [AK1] and the new formulation of the Teichmüller TQFT [AK2] for the one-vertex H-triangulation of $(S^3,6_1)$. We show that the two formulations give equivalent answers. Furthermore we apply a formal stationary phase analysis and arrive at the Andersen- Kashaev volume conjecture as stated in [AK1, Conj. 1]. Furthermore we calculate the first examples of knot complements in the new formulation showing that the new formulat…
We obtain bounds on the numbers of intersections between triangulations as the conformal structure of a surface varies along a Teichm{ü}ller geodesic contained in an $\mathrm{SL}\left(2,\mathbb{R}\right)$-orbit closure of rank 1 in the moduli space of Abelian differentials. For $0 \leq θ\leq 1$, we obtain an exponential bound on the number of closed geodesics in the orbit closure, of length at most $R$, that spend at least $θ$-fraction of their length in a region with short saddle connections.
We produce a one-parameter family of coordinates $\{Ψ_h\}_{h\in\mathbb{R}}$ of the decorated Teichmüller space of an ideally triangulated punctured surface $(S,T)$ with negative Euler characteristic, which is a deformation of Penner's simplicial coordinate \cite{P1}. If $h\geqslant0$, the decorated Teichmüller space in the $Ψ_h$ coordinate becomes an explicit convex polytope $P(T)$ independent of $h$, and if $h<0$, the decorated Teichmüller space becomes an explicit bounded convex polytope $P_h(T)$ so that $P_h(T)\subset P_{h'}(T)$ if $h<h'$. As a consequence, Bowditch-Epstein and Penner's cel…
We define an associative algebra AS_h(S) generated by framed arcs and links over a punctured surface S which is a quantization of the Poisson algebra C(S) of arcs and curves on S. We then construct a Poisson algebra homomorphism from C(S) to the space of smooth functions on the decorated Teichmuller space endowed with the Weil-Petersson Poisson structure. The construction relies on a collection of geodesic lengths identities in hyperbolic geometry which generalize Penner's Ptolemy relation, the trace identities and Wolpert's cosine formula. As a consequence, we derive an explicit formula for t…
The space of broken hyperbolic structures generalizes the Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations (equivalently, the space of projectivized affine foliations) generalizes the space of projectivized measured foliations. Just as projectivized measured foliations provide Thurston's boundary for Teichmüller space, so too do projectivized broken measured foliations form a boundary for the space of broken hyperbolic structures. In this paper, we naturally extend the Weil-Petersson Kähler two-form and the Thurston symplectic form to their br…
The Maskit embedding M of a surface Σis the space of geometrically finite groups on the boundary of quasifuchsian space for which the `top' end is homeomorphic to Σ, while the `bottom' end consists of two triply punctured spheres, the remains of Σwhen two fixed disjoint curves have been pinched. As such representations vary in the character variety, the conformal structure on the top side varies over the Teichmüller space T(Σ). We investigate M when Σis a twice punctured torus, using the method of pleating rays. Fix a projective measure class [μ] supported on closed curves on Σ. The pleating r…
Let $S$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $ω$ be a pseudo-Anosov map of $S$ that is obtained from Dehn twists along two families $\{A,B\}$ of simple closed geodesics that fill $S$. Then $ω$ can be realized as an extremal Teichmüller mapping on a surface of type $(p,n)$ which is also denoted by $S$. Let $ϕ$ be the corresponding holomorphic quadratic differential on $S$. In this paper, we compare the locations of some distinguished points on $S$ in the $ϕ$-flat metric to their locations with respect to the complete hyperbolic metric. More precisely, we show that all possi…
Based on earlier work of the latter two named authors on the higher super-Teichmueller space with $\mathcal{N}=1$, a component of the flat $OSp(1|2)$ connections on a punctured surface, here we extend to the case $\mathcal{N}=2$ of flat $OSp(2|2)$ connections. Indeed, we construct here coordinates on the higher super-Teichmueller space of a surface $F$ with at least one puncture associated to the supergroup $OSp(2|2)$, which in particular specializes to give another treatment for $\mathcal{N}=1$ simpler than the earlier work. The Minkowski space in the current case, where the corresponding sup…
We consider harmonic diffeomorphisms to a fixed hyperbolic target $Y$, from a family of domain Riemann surfaces degenerating along a Teichmüller ray. We use the work of Minsky to show that there is a limiting harmonic map from the conformal limit of the Teichmüller ray, to a crowned hyperbolic surface. The target surface is the metric completion of the complement of a geodesic lamination on $Y$. The conformal limit is obtained by attaching half-planes and cylinders to the critical graph of the holomorphic quadratic differential determining the ray. As an application, we provide a new proof of …
