Geometry and Dynamics in the Teichmüller space and the Outer space.

我们建议研究泰希米勒空间和密切相关的外层空间的几何和动力学的各个方面。下面是我的活跃项目列表以及参与每个项目的合作者的姓名。 1--配备Thurston度量的Teichmüller空间的几何（与大卫大仲马，Babak Modami，安娜Lenzhen，Jing Tao和Fanny卡塞尔） 瑟斯顿介绍了一个度量的泰希米勒空间，使用双曲几何，而不是共形几何，以确定两点之间的距离。这个度规在许多方面都更自然，特别是它与泰希米勒空间的瑟斯顿边界相互作用的方式。最近的研究表明，它使Teichmüller空间具有丰富的结构。然而，它的几何形状仍然较大未经检查。 2--Teichmüller度量空间的几何（与Maxime Fortier Bourque，Misha Kapovich和Robert Young合著） 这是一个古老的话题，但有许多开放的问题。我们研究Teichmüller空间的凸性以及Dehn函数在Teichmüller空间中的行为。 3-有限和无限型平移曲面（与Anja Randecker和霍华德Masur合著） 这些问题是平面领域最近进展的自然延伸（包括Eskin-Mirzakhani的工作）。我们研究的方向在一个翻译表面的无限型定义一个独特的遍历叶理，同样的问题问维奇的情况下，表面的有限型。 4-映射中的随机游走（与Alex Eskin） Teichmüller空间边界上的Lebesgue测度能作为映射类群中随机游动的平稳测度吗？ 5-外层空间几何学及相关复合体（与Mladen Bestelf和Yulan Qing合著） 外层空间是一个与泰希米勒空间直接类比的空间。然而，可用的工具却少得多。例如，是否存在类似于Out（F_n）的距离公式？I自由因子复均匀双曲？自由分裂复形是一致双曲的吗？ 6--Teichmüller空间中的计数问题（与Juan Souto） 这是对Mirzakhani工作的延续和延伸。考虑点在Teichmüller空间作为测地线电流提供了一个新的观点，其中，使用类比的对称空间，许多困难的计数问题变得平易近人。 7-大地图类组（与朱丽叶朱丽叶Bavard和斯宾塞Dowdall） 这是一个新的领域，甚至最基本的问题都是开放的。我们问这些大的映射类群是否具有Schreier意义下的强畸变性质。 8-模空间的形状（与Maxime Fortier Bourque和Robert Young） 模空间的形状仍然是个谜。例如，什么是模空间的Cheeger常数？是不是到处都是粗糙的、同质的？它是否类似于一个扩展图？

We propose to study various aspects of geometry and dynamics of Teichmüller space and the closely related Outer space. Below is the list of my active projects and the names of collaborators that are involved in each of these projects. 1-- Geometry of Teichmüller space equipped with Thurston Metric (with David Dumas, Babak Modami, Anna Lenzhen, Jing Tao and Fanny Kassel) Thurston introduced a metric on Teichmüller space that uses the hyperbolic geometry, rather than conformal geometry, to define the distance between two points. This metric is more natural in many ways, in particular, in the way it interacts with the Thurston boundary of Teichmüller space. Recent studies have shown that it equips Teichmüller space with rich structure. However, its geometry remains larger unexamined. 2-- Geometry of Teichmüller Space Equipped with Teichmüller Metric (with Maxime Fortier Bourque, Misha Kapovich and Robert Young) This is an old topic, however with many open problems. We examine convexity properties of Teichmüller space as well as the behavior of Dehn functions in Teichmüller space. 3- Translation Surfaces of Finite and Infinite Type (with Anja Randecker and Howard Masur) These problems are natural extensions of recent progress in the field of flat surfaces (including the work of Eskin-Mirzakhani). We examine which directions in a translation surface of infinite type define a uniquely ergodic foliation, the same question asked by Veech in the case of surfaces of finite type. 4- Random Walks in Mapping class group (with Alex Eskin) Can the Lebesgue measure in the boundary of Teichmüller space be obtained as the stationary measure of a random walk in the mapping class group? 5- Geometry of Outer space and related complexes (with Mladen Bestvina and Yulan Qing) Outer space is a space constructed as a direct analogy with Teichmüller space. However, there are considerably fewer tools available. For example, is there an analogue of the distance formula for Out(F_n)? I the free factor complex uniformly hyperbolic? Is the free splitting complex uniformly hyperbolic? 6-- Counting problems in Teichmüller space (with Juan Souto) This is following and extending the work of Mirzakhani. Considering points in Teichmüller space as geodesic currents provides a new point of view where, using analogies with the symmetric space, many difficult counting problems become approachable. 7- Big Mapping Class Group (with Juliette Juliette Bavard and Spencer Dowdall) This is a new field and even the most basic problems are open. We ask if these large mapping class group have the strong distortion property in the sense of Schreier. 8- Shape of Moduli Space (with Maxime Fortier Bourque and Robert Young) The shape of moduli space remains mysterious. For example, what is the Cheeger constant of moduli space? Is it coarse, homogenous almost everywhere? Does it resemble an expander graph?