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

这个项目的目的是将群作用的性质与齐次空间、欧几里得建筑和模空间联系起来。具体研究内容包括：（1）半单李群的离散子群的拟等距。(2)函数域算术群的单调性（与Kai-Uwe Bux）。(3)德恩函数的算术群（与格雷戈里马古利斯）。(4)阿贝尔微分空间上的幂单流（与Kariane Calta）。齐性空间是几何对象，可用于分析满足某些方程的数组。欧几里得建筑物也是几何对象，它们可以用来研究广义“数”的数组。模空间是几何对象，可用于参数化与复数方程相关的空间。许多努力已经投入到比较和对比这三个对象，因为每一个都在数学领域发挥着重要作用。本项目的目标是继续研究这三个基本对象之间的联系，希望对三者之间关系的更深入理解将为每个人带来见解。

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.