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

In the paper's words

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. [...]

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The displacement-to-distance ratio at the barycenter is $\sqrt{2n/(n-1)}$, decreasing from $2 (n = 2)$ through $\sqrt{3} \approx 1.732 (n = 3)$ toward $\sqrt{2}$ as $n \to \infty$.

   from 4. Analysis

2. Quantity General $p = 3$ Source Ball vertices V(2) $1 + (p+1)(p^{2}-1)/(p-1)$ 17 §4.1 Ball edges E(2) V(2) − 1 16 §4.1 Ball vertices V(3) $1 + (p+1)(p^{3}-1)/(p-1)$ 53 §4.1 Ball edges E(3) V(3) − 1 52 §4.1 Vertex finite layer $p(p^{2}-1)$ 24 §4.2 Generic-point finite layer p(p−1) 6 §4.2 [Stab(v₀): Stab(e)] p+1 4 §4.3 [Stab(v₀): $I_{1}]$ $p^{2}-1$ 8 §4.3 Covering radius of $X^{0}$ 1/2 1/2 §4.5 Displacement gap $\delta (x) = 2\cdot d(x, X^{0})$ — §4.5 Vertex-rigidity bound $r \lt 1/2$ forces vertex support — §4.6 Fusion channels per vertex / generic point p+1 / 1 4 / 1 §4.7 R1 (Skeleton efficiency).

   from 5. Results

3. The vertex set meets every chamber and is minimal with this property; edges = vertices − 1 in every finite subtree ($52/53 \approx 0.981$ edge-to-vertex ratio at $N = 3$, tending to 1).

   from 5. Results

4. For all SL_n, the chamber-barycenter displacement is exactly 1 and the displacement-to-distance ratio is $\sqrt{2n/(n-1)} \gt \sqrt{2}$, under the stated worst-case assumption (§4.9).

   from 5. Results

5. (2) Generator bound vs. exact count: the edge count E(r) is an upper bound on elementary braid generators in the sense of [12]; braid relations may reduce the minimal generating set, so the bound is not an exact count of independent generators.

   from 6. Discussion

6. The vertex set is the minimal G-stable skeleton meeting every chamber (17 vertices/16 edges in the $p = 3, r = 2$ ball; 53/52 at $r = 3$); it is the unique orbit whose stabilizer surjects onto the full finite group $\mathrm{S}\mathrm{L}_{2}(\mathbb{F}_{p})$ of order $p(p^{2}-1) = 24$ at $p = 3$, providing $p+1 = 4$ fusion channels per vertex; generic points cost a stabilizer factor of p+1, collapse the finite layer to the Borel of order 6, and supply exactly one channel and no new braid generators.

   from 7. Conclusion

How it is done

1. 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 ($|\mathrm{S}\mathrm{L}_{2}(\mathbb{F}_{p})| = p(p^{2}-1)$, edge-stabilizer index p+1, pro-p Iwahori filtration index $p^{2}-1$), 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 $r \lt 1/2$ is vertex-supported.

   from Abstract

2. In a related lattice-theoretic register, [3] proves a compression-of-angles theorem for buildings of type A: for the set Lat(V) of lattices in a p-adic vector space, the complex distance is a complete system of invariants for pairs under the full linear group, and a Nazarov-semigroup element compresses configurations toward apartments.

   from 2. Background and Related Work

3. Let $F = \mathbb{Q}_{p}, G = \mathrm{S}\mathrm{L}_{2}(\mathrm{F})$, and let X be the Bruhat–Tits building of G, the (p+1)-regular tree.

   from 3. Methods

4. By (I1), the sphere of radius $n \geq 1$ has size $|S_{n}| = (p+1)p^{n-1}$ (each vertex at distance n−1 has p forward edges, one edge returning).

   from 4. Analysis

What would falsify it

1. Limitations (rank one, finite radius, projected higher-rank bounds) and falsification criteria are stated explicitly.

   from Abstract

2. Third, we state honest limitations and falsification criteria (Section 6).

   from 1. Introduction

3. A construction of a p-adic anyon model in the sense of [13] whose fusion ring is faithfully realized by the stabilizer of a generic edge point (a Borel-type subgroup) would falsify R2–R5.

   from 6. Discussion

Where the claim stops

1. What the results do and do not show.

   from 6. Discussion

2. This is a mechanism, not a theorem about the QNFO constructions: [12],[13],[14] define their structures on vertices by choice, and our results explain why that choice is natural — indeed nearly forced — but do not prove that alternative point-based constructions are impossible.

   from 6. Discussion

3. (3) Boundary behavior: in the non-split setting of [5], limit points may carry essential data that vertices alone do not; our analysis does not address boundary behavior.

   from 6. Discussion

Where to read

1. Abstract253 words · 1 min
2. 1. Introduction305 words · 1 min
3. 2. Background and Related Work689 words · 3 min
4. 3. Methods262 words · 1 min
5. 4. Analysis997 words · 4 min
6. 5. Results314 words · 1 min
7. 6. Discussion414 words · 2 min
8. 7. Conclusion183 words · 1 min
9. Appendix A. Divergence report436 words · 2 min
10. Appendix B. Claim attribution300 words · 1 min