Speaker: Sanghoon Kwon
Affiliation: Catholic Kwandong University, Republic of Korea
Title of Talk: Geometry and dynamics of group actions on Bruhat–Tits buildings
Date: September 12, 2025
Time: 11:00:00 Hours
Venue: AG-77

Abstract: Bruhat–Tits buildings provide a rich geometric framework for studying linear groups over non-Archimedean local fields. In this lecture, we will introduce the Bruhat–Tits tree associated with PGL2 as a motivating example, highlighting its combinatorial and metric properties, and then discuss the general structure of higher-dimensional buildings. We will explore how these spaces serve as analogues of symmetric spaces in the non-Archimedean setting, illustrating their role as natural domains for group actions. From the dynamical perspective, we will examine the action of arithmetic lattices and diagonal subgroups on these spaces, drawing parallels with familiar phenomena from the theory of Lie groups over real numbers. Topics will include orbits and stabilizers, compactness and reduction theory, and basic spectral aspects of the Laplacian on trees. Along the way, we will emphasize applications to number theory and geometry, such as connections to Diophantine approximation, random walks, and representation theory. The lecture is designed for graduate students; no prior knowledge of buildings is assumed. The aim is to convey the intuition behind the geometry of Bruhat–Tits spaces and to showcase how dynamics of group actions on them opens a gateway between geometry, algebra, and number theory. I will also present several simple exercises along with some open problems.