Why Vertices, Not Points? Vertex-Anchored Braiding, Stabilizer Arithmetic, and Displacement Gaps on Bruhat–Tits Buildings

Abstract

Bruhat–Tits buildings are the canonical combinatorial-geometric objects attached to reductive groups over non-Archimedean local fields, and the recent QNFO constructions of p-adic braid groups [12], p-adic anyon models [13], and a p-adic Temperley–Lieb parameter [14] anchor their discrete braiding data at the *vertex set* of the building rather than at arbitrary points of its geometric realization. This paper answers the question posed in the title with a combination of exact arithmetic and explicit metric computation in the rank-one case, where the building of SL₂ over ℚ_p is a (p+1)-regular tree. We compute ball growth exactly (for p = 3: a radius-2 ball contains 17 vertices and 16 edges; a radius-3 ball contains 53 vertices and 52 edges), we compute the finite stabilizer layers exactly (|SL₂(𝔽_p)| = p(p²−1), edge-stabilizer index p+1, pro-p Iwahori filtration index p²−1), and we compute a displacement gap: a point at distance d from the nearest vertex is moved by exactly 2d by the Weyl element of the vertex stabilizer, so the covering radius of the vertex set is 1/2 and any vertex-stabilizer-equivariant structure supported in a ball of radius r < 1/2 is vertex-supported. We show that the vertex set is the unique G-orbit whose stabilizer surjects onto the full finite group SL₂(𝔽_p), yielding p+1 fusion channels per vertex, while generic edge points cost a stabilizer factor of p+1, collapse the finite layer to the Borel of order p(p−1), and supply no new braid generators. Limitations (rank one, finite radius, projected higher-rank bounds) and falsification criteria are stated explicitly. ## 1. Introduction Let F be a non-Archimedean local field (e.g. ℚ_p) and G a connected reductive F-group. The Bruhat–Tits building X = X(G, F) is a polysimplicial complex on which G(F) acts by isometries; its apartments are Coxeter complexes, its maximal simplices correspond to Iwahori subgroups, and its vertices correspond to parahoric subgroups — compact open subgroups that p Full text and updates: papers.qnfo.org/papers/why-vertices-not-points-vertex-anchored-braiding-stabilizer-arithmetic-and-displ/