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

我们建议研究Teichmüler空间和与之密切相关的外层空间的几何和动力学的各个方面。下面是我正在进行的项目的列表以及参与每个项目的合作者的姓名。 1--配备瑟斯顿度规的泰希米勒空间的几何(与大卫·大仲马、巴布克·莫达米、安娜·伦珍、景涛和范妮·卡塞尔) 瑟斯顿在Teichmüler空间上引入了一种度量，它使用双曲几何而不是保角几何来定义两点之间的距离。这个度规在许多方面更自然，特别是它与Teichmüler空间的瑟斯顿边界相互作用的方式。最近的研究表明，它使Teichmüler空间具有丰富的结构。然而，它的几何形状仍然更大，还没有得到检验。 2--配备Teichmüler度量的Teichmüler空间的几何(与Maxime Fortier Bourque，Misha Kapovich和Robert Young) 这是一个古老的话题，但也有许多悬而未决的问题。我们研究了Teichmüler空间的凸性以及Dehn函数在Teichmüler空间中的行为。 3-有限和无限型平移曲面(Anja Randecker和Howard Masur) 这些问题是平面领域最新进展(包括Ekin-Mirzakhani的工作)的自然延伸。我们研究了无限类型的平移曲面中的哪些方向定义了唯一的遍历叶层，这与Veech在有限类型曲面的情况下提出的问题相同。 4-在映射类组中随机行走(与Alex Ekin一起) 作为映射类群中随机游动的平稳测度，Teichmüler空间边界上的勒贝格测度可以得到吗？ 5-外层空间几何和相关复合体(与Mladen Bestvina和玉兰青) 外层空间是一种直接类比于泰希米勒空间的空间。然而，可用的工具要少得多。例如，有没有类似于Out(F_N)的距离公式？I自由因子复数一致双曲线？自由分裂复形是一致双曲线的吗？ 6--Teichmüler空间中的计数问题(与Juan Souto) 这是对米尔扎哈尼工作的延续和延伸。将Teichmüler空间中的点视为测地线电流提供了一种新的观点，通过与对称空间的类比，许多困难的计数问题变得容易处理。 7-大型地图班组(朱丽叶·朱丽叶·巴沃德和斯宾塞·道德尔) 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意义下的强扭曲性质。 模空间的8-形状(与Maxime Fortier Bourque和Robert Young) 模数空间的形状仍然是个谜。例如，模空间的切格常数是多少？它是不是几乎到处都是粗糙、均匀的？它类似于扩展图吗？

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?