Distance formulas in Bruhat-Tits building of $\mathrm{SL}_d(\mathbb{Q}_p)$

TL;DR

This paper derives explicit combinatorial formulas for distances in the Bruhat-Tits building of (\u211a_p), including a general distance formula and a minimal total distance to a set, with special cases analyzed.

Contribution

It introduces explicit formulas for distances in the Bruhat-Tits building of (_p), extending previous understanding without needing common apartments.

Findings

Explicit formula for graph distance between vertices.

Formula for minimal total distance to a set of vertices.

Special case analysis for (_p) in the appendix.

Abstract

We study the distance on the Bruhat-Tits building of the group $\mathrm{SL}_d(\mathbb{Q}_p)$ (and its other combinatorial properties). Coding its vertices by certain matrix representatives, we introduce a way how to build formulas with combinatorial meanings. In Theorem 1, we give an explicit formula for the graph distance $\delta(\alpha,\beta)$ of two vertices $\alpha$ and $\beta$ (without having to specify their common apartment).Our main result, Theorem 2, then extends the distance formula to a formula for the smallest total distance of a vertex from a given finite set of vertices. In the appendix we consider the case of $\mathrm{SL}_2(\mathbb{Q}_p)$ and give a formula for the number of edges shared by two given apartments.