Alexandra Pettet, Juan Souto
Let T_n be the Teichmueller space of flat metrics on the n-dimensional torus and identify SL(n,Z) with the corresponding mapping class group. We prove that the subset Y consisting of those points at which the systoles generate the fundamental group of the torus is, for n > 4, not contractible. In particular, Y is not an SL(n,Z)-equivariant deformation retract of T_n.
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Estimating the dimension of Thurston spine
本文研究了 Thurston 脊 Pg 的余维数性质，证明了存在无限整数集 A 使得当亏格 g 在 A 中变化时 codim Pg 为 o(g/ log g)，从而验证了 M. Fortier Bourque 的猜想。
We prove that there are Fenchel-Nielsen coordinates for the Teichmueller space of a finite area hyperbolic surface with respect to which the length functions are convex.
We define and study metrics and weak metrics on the Teichmueller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of properly embedded arcs in the surface. We give a comparison between the defined metrics on regions of Teichmueller space which we call $\varepsilon_0$-relative $ε$-thick parts} for $ε>0$ and $\varepsilon_0\geq ε>0$.
We consider some metrics and weak metrics defined on the Teichmueller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call "$ε_0$-relative $ε$-thick parts", and whose definition depends on the choice of some positive constants $ε_0$ and $ε$. Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal l…
We characterize which cobounded quasigeodesics in the Teichmueller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path gamma in T, we show that gamma is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canonical hyperbolic plane bundle over gamma is a hyperbolic metric space. As an application, for complete hyperbolic 3-manifolds N with finitely generated, freely indecomposable fundamental group and with bounded geometry, we give a new construction of model geometries for the geometrically inf…
Usually, the description of tangent spaces to the Teichmueller space $\mathscr{T}(Σ_{g})$ of a compact Riemann surface $Σ_{g}$ of genus $g \geq 2$ (which we can identify with the quotient space $\mathbb{H}^{2} / Γ_{g}$ of the upper half plane $\mathbb{H}^{2}$ by a discrete cocompact subgroup $Γ_{g}$ of $\mathrm{PSL}(2, \mathbb{R})$) comes in two different flavours: the space of holomorphic quadratic differentials on $Σ_{g}$ which are holomorphic sections of the tensor square of the canonical line bundle of $Σ_{g}$ and the first cohomology group $H^{1}(Γ_{g}; \mathfrak{g})$ of the fundamental g…
Let Q be a connected component of a stratum in the space of quadratic differentials for a non-exceptional Riemann surface of finite type. We show that the probability measure on Q in the Lebesgue measure class which is invariant under the Teichmueller flow is obtained by Bowen's construction.
Given two measured laminations mu and nu in a hyperbolic surface which fill up the surface, Kerckhoff [Lines of Minima in Teichmueller space, Duke Math J. 65 (1992) 187-213] defines an associated line of minima along which convex combinations of the length functions of mu and nu are minimised. This is a line in Teichmueller space which can be thought as analogous to the geodesic in hyperbolic space determined by two points at infinity. We show that when mu is uniquely ergodic, this line converges to the projective lamination [mu], but that when mu is rational, the line converges not to [mu], b…
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…
Let S be a nonexceptional oriented surface of finite type. We construct an uncountable family of probability measures on the space of area on holomorphic quadratic differentials over the moduli space for S containing the usual Lebesgue measure. These measures are invariant under the Teichmueller geodesic flow, and they are mixing, absolutely continuous with respect to the stable and unstable foliation adn exponentially recurrent to a compact set. Finally we show that the critical exponent of the mapping class group equals the dimension of the Teichmueller space for S. Moreover, this critical e…
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…
In this work, we study the asymptotic geometry of the mapping class group and Teichmueller space. We introduce tools for analyzing the geometry of `projection' maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several applications of this analysis. One of which is that the asymptotic cone of the mapping class group of any surface is tree-graded in the sense of Drutu and Sapir; this tree-grading has several consequences including answering a question of Drutu and Sapir concerning relatively hype…
Universal Spin Teichmueller Theory, I. The action of P(SL(2,Z)) on Tess^+
本文引入旋量映射类群P(SL(2,Z))与旋量通用泰希米勒空间Tess^+，证明三个组合定义的元素可生成该群，为带旋量结构的有限型双曲曲面提供通用框架，配套论文给出其有限表现。
Symbolic dynamics for the Teichmueller flow
本文通过构造有限型子移位及其Borel悬浮，建立到Teichmüller流的有限对一半共轭，从而证明分层分支上的Lebesgue测度是唯一最大熵测度。
From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these determine the same set of Teichmueller curves as the non-obtuse lattice triangles which were classified by Kenyon, Smillie, and Puchta. We also identify a pseudo-Anosov automorphism whose dilatation is Lehmer's number, and show that this is minimal for the groups under considerati…
Universal Spin Teichmueller Theory, II. Finite Presentation of P(SL(2,Z))
该研究定义了旋量映射类群P(SL(2,Z))，通过计算旋量覆叠的轨形甲板群得到其有限表示，还发现对易关系与根格的关联并推测其自同构群为散在群，配套论文验证了群的拓扑对应性。
We construct infinitely many smooth 4-manifolds which are homotopy equivalent to $S^2$ but do not admit a spine, i.e., a piecewise-linear embedding of $S^2$ which realizes the homotopy equivalence. This is the remaining case in the existence problem for codimension-2 spines in simply-connected manifolds. The obstruction comes from the Heegaard Floer $d$ invariants.
The Thurston spine and equivariant cell decompositions in genus 2
本文构造了亏格 2 Teichmüller 空间的 Thurston 脊柱，识别了所有填充 systole 集合的映射类群轨道，并与基于 Rivin 角坐标的胞腔分解进行了对比。
Geometrical spines are defined for 3-manifolds with natural metrics, in particular, for lens manifolds. We show that any spine of L(p,q) close enough to its geometrical spine (i.e., to the cut locus with respect to the standard metric) contains at least E(p,q)-3 vertices, which is exactly the conjectured value for Matveev's complexity of L(p,q); here E(p,q) stands for the sum of the elements of the continued fraction expansion of p/q. As a byproduct, we find the minimal (over all triangulations) rotation distance (the term coined by Sleator, Tarjan, and Thurston) between a triangulation of a r…
An equivariant deformation retraction of the Thurston spine
本文证明了闭可定向曲面 Teichmüller 空间存在映射类群等变的形变收缩到 Thurston 骨架，其维数等于映射类群的虚上同调维数。
A spine for the decorated Teichmüller space of a punctured non-orientable surface
本文为带穿孔非可定向曲面的装饰 Teichmüller 空间构造骨架，证明形变收缩关于纯映射类群等变，并给出单穿孔情形下最小维数的真作用分类空间模型。
Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for all N. Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit canonical 1-cocycle of the Teichmueller space. The main tool for these results is the …
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…
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