Simplicial volumes in Bruhat-Tits buildings of split classical type

TL;DR

This paper investigates the asymptotic growth of simplicial volumes in Bruhat-Tits buildings of split classical types, deriving formulas and dominant terms using concave functions and q-exponential polynomial theory.

Contribution

It introduces a new approach to compute and analyze the asymptotic behavior of simplicial volumes in these buildings, connecting geometric and algebraic methods.

Findings

Derived a formula for simplicial volume using concave functions

Identified the dominant asymptotic term with q-exponential polynomial theory

Provided insights into the growth rate of vertex counts within simplicial distances

Abstract

In a Bruhat-Tits building of split classical type (that is, of type $A_n$, $B_n$, $C_n$, $D_n$, and any combination of them) over a local field, the simplicial volume counts the vertices within the given simplicial distance from a special vertex. This paper aims to study the asymptotic growth of the simplicial volume. A formula of the simplicial volume is deduced from the theory of concave functions. Then the dominant term in its asymptotic growth is found using the theory of $q$-exponential polynomials developed in this paper.