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

该项目旨在探讨同质空间、欧几里得建筑和模空间中群体行为的相关属性。具体的研究课题包括：(1)半单李群的离散子群的拟等距。(2)函数域算术群的有限性（带Kai-Uwe Bux）。(3)算术群的Dehn函数（与Gregory Margulis合作）。(4)阿贝尔微分空间上的幂偶流（带Kariane Calta）。齐次空间是可以用来分析满足某些方程的数字数组的几何对象。欧几里得建筑也是几何对象，它们可以用来研究广义“数”的数组。模空间是一种几何对象，可以用来参数化与复数方程相关的空间。对这三个对象进行比较和对比已经付出了很多努力，因为它们在数学领域中都起着重要的作用。该项目的目标是继续研究这三个基本对象之间的联系，希望对三者之间的关系有更深入的了解，从而为每个个体提供见解。

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.