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

泰希米勒空间和外层空间的几何和动力学。

• 批准号: RGPIN-2018-06486
• 负责人: Rafi, Kasra
• 金额: $ 1.68万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758694
• 关键词: Geometry; Dynamics; Teichm; ller; space

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?

我们建议研究的几何和动力学的各个方面的teichm<s:1>勒空间和密切相关的外层空间。下面是我正在进行的项目列表，以及参与这些项目的合作者的名字。1-配备Thurston度量的teichm<e:1>空间的几何（与David Dumas， Babak Modami, Anna Lenzhen， Jing Tao和Fanny Kassel） Thurston在teichm<e:1>空间上引入了一个度量，该度量使用双曲几何而不是共形几何来定义两点之间的距离。这个度规在很多方面更自然，特别是它与teichm<s:1> ller空间的Thurston边界相互作用的方式。近年来的研究表明，它使teichm<e:1>空间具有丰富的结构。然而，它的几何形状仍有待进一步研究。2-配备teichm<e:1> ller度量的teichm<e:1> ller空间的几何（与Maxime Fortier Bourque， Misha Kapovich和Robert Young）这是一个古老的话题，但是有许多开放的问题。我们研究了teichm<e:1> ller空间的凸性以及Dehn函数在teichm<e:1> ller空间中的行为。有限型和无限型的平移曲面（与Anja Randecker和Howard Masur）这些问题是平面领域最新进展（包括Eskin-Mirzakhani的工作）的自然延伸。我们考察了无限型平移曲面上哪些方向定义了唯一遍历叶理，这与Veech在有限型曲面上提出的问题相同。4-映射类群中的随机漫步（with Alex Eskin）能否得到teichm<e:1>空间边界上的Lebesgue测度作为映射类群中随机漫步的平稳测度？5 .外空间几何及其相关复合体（与Mladen Bestvina和Yulan Qing合作）外空间是与teichm<s:1> ller空间直接类比而构建的空间。然而，可用的工具要少得多。例如，对于Out（F_n）是否有类似的距离公式？自由因子是均匀双曲的吗？自由分裂复合体是均匀双曲的吗？6- teichm<e:1>空间中的计数问题（与Juan Souto合作）这是对Mirzakhani工作的继承和扩展。将teichmller空间中的点视为测地线电流提供了一种新的观点，在这种观点中，使用与对称空间的类比，许多困难的计数问题变得可以接近。这是一个新的领域，甚至最基本的问题都是开放的。8-模空间的形状（与Maxime Fortier Bourque和Robert Young合著）模空间的形状仍然是一个谜。例如，模空间的Cheeger常数是多少？它是粗糙的，几乎到处都是同质的吗？它像一个展开图吗？