Abstract
We study the distance on the Bruhat–Tits building of the group [Formula: see text] (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 [Formula: see text] of two vertices [Formula: see text] and [Formula: see text] (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 [Formula: see text] and give a formula for the number of edges shared by two given apartments.
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