论文标题
通过CAT（0）Cube Complextes映射类组和Teichmüller空间中的通用几何形状
The geometry of genericity in mapping class groups and Teichmüller spaces via CAT(0) cube complexes
论文作者
论文摘要
在具有双曲线特性的空间上随机行走倾向于跨线性跟踪地球射线，该射线指向某些双曲线样方向。清 - -tiozzo最近引入了均匀的摩尔斯山边界，并证明了该边界是一种准偶然的不变性，它在广泛的环境中捕获了通用方向的概念。 在本文中，我们开发了在映射类组和Teichmüller空间中sublinear Morseness的几何基础。我们证明它们的跨性摩尔斯边界是可见性空间，并接纳对曲线图边界的连续地注射。此外，我们完全根据这些空间的层次结构来表征均方根。 我们的技术包括开发通过CAT（0）Cube Complexs在分层双曲线空间中中位射线壳建模的工具。该分析的一部分涉及建立曲线图的几何形状与近似立方体复合物中超平面的组合之间的直接连接。
Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved that this boundary is a quasi-isometry invariant which captures a notion of generic direction in a broad context. In this article, we develop the geometric foundations of sublinear Morseness in the mapping class group and Teichmüller space. We prove that their sublinearly Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the curve graph. Moreover, we completely characterize sublinear Morseness in terms of the hierarchical structures of these spaces. Our techniques include developing tools for modeling the hulls of median rays in hierarchically hyperbolic spaces via CAT(0) cube complexes. Part of this analysis involves establishing direct connections between the geometry of the curve graph and the combinatorics of hyperplanes in the approximating cube complexes.
