Group actions on homogeneous spaces, Euclidean buildings, and moduli spaces

本课题主要研究齐次空间、欧几里得建筑和模空间上的群作用的性质。具体的研究内容包括：(1)半单李群的离散子群的拟等距。(2)函数域算术群(带Kai-Uwe Bux)的有限性。(3)算术群的Dehn函数(与Gregory Marguis)。(4)交换微分空间上的幂等流(具有Kariane Calta)。齐次空间是可以用来分析满足某些方程的数组的几何对象。欧几里得建筑也是几何对象，它们可以用来研究广义类型的“数”的阵列。模空间是几何对象，可用于将与复数方程相关的空间参数化。由于这三个对象在数学领域中都扮演着重要的角色，人们花了很多精力来比较和对比这三个对象。这个项目的目标是继续考察这三个基本对象之间的联系，希望对这三个对象之间的关系有更深入的了解，这将对每个人都有深刻的见解。

This project is aimed at relating properties of group actions onhomogeneous spaces, Euclidean buildings, and moduli spaces. Specifictopics for investigation include: (1) Quasi-isometries of discretesubgroups of semisimple Lie groups. (2) Finiteness properties offunction-field-arithmetic groups (with Kai-Uwe Bux). (3) Dehn functions ofarithmetic groups (with Gregory Margulis). (4) Unipotent flows on spacesof abelian differentials (with Kariane Calta).Homogeneous spaces are geometric objects that can be used to analyze arrays of numbers that satisfy certain equations. Euclidean buildings are also geometric objects, and they can be used to study arrays of generalized kinds of ``numbers''. Moduli spaces are geometric objects that can be used to parameterize spaces related to equations of complex numbers. Much effort has been put into comparing and contrasting these three objects, as each plays a significant role in the field of mathematics. The goal of this project is to continue to examine links between these three fundamental objects with the hope that a deeper understanding of the relationships between the three will bear insights for each as individuals.