p-Adic Braid Groups on Bruhat-Tits Buildings

In the paper's words

Constructs p-adic braid groups on Bruhat-Tits buildings, establishing the geometry of discrete braiding.

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

Numbers it reports

No sentence of this paper reports a figure with a unit or a quantity word, so none is shown.

How it is done

1. By Ostrowski's theorem, the archimedean absolute value $|\cdot|_\infty$ is only one of infinitely many inequivalent completions of $\mathbb{Q}$.

   from Abstract

2. By Ostrowski's theorem, $\mathbb{Q}$ admits exactly two types of non-trivial absolute values up to equivalence: the archimedean $|\cdot|_\infty$ and the p-adic $|\cdot|_p$ for each prime $p$.

   from 1.2 Motivation: Why p-Adic?

3. The proof reduces to verifying that the three-vertex configuration on a tree admits exactly two homotopically distinct exchanges (clockwise and counterclockwise around the unique branch point), and these satisfy the braid relation identically.

   from 4.3 Proof of the Braid Relation $\sigma_1\sigma_2\sigma_1 = \sigma_2\sigma_1\sigma_2$

4. Their construction uses the algebraic inverse limit over finite quotients — a purely algebraic approach without geometric building data.

   from 5. Connection to Existing p-Adic Framed Braids

What would falsify it

The paper's text states no explicit falsification criterion; the limits below are what it says about where the claim stops.

Where the claim stops

1. The classical definition $B_n \cong \pi_1(\text{Conf}_n(\mathbb{R}^2))$ fails entirely.

   from 1.3 The Challenge: No Continuous Paths in $\mathbb{Q}_p$

2. However, the key insight is that the ultrametric nature of the tree means there are no "infinitesimal" braiding operations: any exchange of two particles traverses at least one edge.

   from 3.2 Why Vertices, Not Points?

3. The geometric interpretation, however, differs fundamentally from the archimedean case.

   from 4.1 Definition via Braid Words

Where to read

1. Abstract155 words · 1 min
2. 1. Introduction326 words · 1 min
3. 2. The Bruhat-Tits Tree for $\text{SL}_2(\mathbb{Q}_p)$309 words · 1 min
4. 3. The p-Adic Configuration Space156 words · 1 min
5. 4. The p-Adic Braid Group $B_n(\mathbb{Q}_p)$638 words · 3 min
6. 5. Connection to Existing p-Adic Framed Braids138 words · 1 min
7. 6. The Temperley-Lieb Connection228 words · 1 min
8. 7. Computational Verification274 words · 1 min
9. 8. Relation to the QNFO Silent Radix Program130 words · 1 min
10. 9. Open Questions165 words · 1 min
11. 10. Conclusion125 words · 1 min