This project is aimed at relating properties of group actions on homogeneous spaces, Euclidean buildings, and moduli spaces. Specific topics for investigation include: (1) Quasi-isometries of discrete subgroups of semisimple Lie groups. (2) Finiteness properties of function-field-arithmetic groups (with Kai-Uwe Bux). (3) Dehn functions of arithmetic groups (with Gregory Margulis). (4) Unipotent flows on spaces of 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.
