UC Santa Cruz

This dissertation studies the notion of simplicial distance on Bruhat-Tits buildings. That is a measure of proximity between vertices in the simplicial structure. The purpose of this research is three-fold: (i). to provide a concrete characterization of the simplicial distance; (ii). to better understand simplicial balls, and (iii). to derive a formula for the simplicial volume and explore its asymptotic growth.To accomplish these goals, a comprehensive examination of vertices becomes necessary. They are analyzed using three frameworks: root systems, norms, and lattices. By leveraging concave functions, we interpret simplicial balls as fixed-point sets of Moy-Prasad subgroups and deduce a formula for the simplicial volume. Additionally, the theory of ?-exponential polynomials is developed to facilitate the asymptotic study. Through this research, we focus on split classical types (namely, types of ??, ??, ??, ??, and any combination thereof) over a local field. The presented findings contribute to the advancement of our understanding of Bruhat-Tits buildings.