Pure braid groups are RFRS

Abstract

Agol in his 2014 ICM proceedings article \cite[Question 11]{Agol14} asks whether braid groups are (virtually) RFRS. We answer this positively by showing that pure braid groups are RFRS. As a consequence, several families of Artin groups are virtually RFRS, including those of type $A_n$ (the braid groups), $B_n=C_n$, $\widetilde A_n$, and $\widetilde C_n$. Our results also provide evidence toward the problem of whether braid groups, and more generally Artin groups, are virtually special; see \cite[Problem 9.4]{HagWi10}, \cite[Problem 13.4]{Wise14}.

Disclosure

“. We thank Ian Agol and Sam Fisher for their helpful communications. We benefited from the assistance of AI tools in our study. In particular, the key idea for the proof of Theorem A was developed from a candidate proof sketch generated by ChatGPT-5.5 pro. 1. Partial RFRS towers In this section, we first recall a way to detect RFRSness using partial RFRS towers from [HWY, §3] then generalize it to the RFRp setting. For a group G, define”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:14:47.332068Z

Structural counts