Derived Pure Braid Groups in the Solvable Filtration

Definitions and mathematical statement

For a positive integer parameter $m$, let $P_m$ denote the pure braid group on $m$ strands, equivalently \[ P_m=\pi_1\!\left(\operatorname{Conf}_m({\mathbb{C}}),b\right),\qquad \operatorname{Conf}_m({\mathbb{C}})=\{(z_1,\ldots,z_m)\in{\mathbb{C}}^m:z_i\ne z_j\text{ whenever }i\ne j\}, \] where $b$ is a fixed base configuration. For a group $G$, define its derived series by \[ G^{(0)}=G,\qquad G^{(r+1)}=[G^{(r)},G^{(r)}]\quad(r\geq0), \] where $[H,H]$ is the subgroup generated by the commutators $xyx^{-1}y^{-1}$ with $x,y\in H$.

The supplied corrected formulation asks whether \[ (P_m)^{(n+2)}\subseteq\mathcal F_n^m \qquad\text{for every integer }n\geq1. \] Here the superscript $(n+2)$ means the $(n+2)$-nd term of the derived series, as specified by the supplied corrected statement. The original source transcription is $ (P_m)^{n+2}\subset F_n^m$ for every $n\geq1$; its unparenthesized exponent is not independently explanatory.

The record's title identifies the intended context as the solvable filtration, but the supplied material does not define $\mathcal F_n^m$, its ambient group, or the identification or homomorphism through which a pure braid is regarded as an element of that ambient group. Consequently, the displayed inclusion records the intended target and requires these missing specifications before it becomes a fully determined mathematical assertion. The record also does not state whether $m$ is fixed or universally quantified, or whether additional restrictions on $m$ apply; no universal quantifier over $m$ is imposed here.

The supplied dated assessment describes only a base containment of the squared pure braid group in the zero-level filtration as established, and describes the higher-derived inclusion as unresolved. The meaning of squared pure braid group is not defined in the record, so this phrase is retained as a qualification rather than converted into a subgroup formula. These are supplied curation claims, not independently verified mathematical or literature conclusions.

Short English statement

The question asks whether the $(n+2)$-nd derived subgroup of the pure braid group lies in level $n$ of the intended solvable filtration for every integer $n\geq1$. The filtration, the ambient interpretation of containment, and the quantification over the strand parameter $m$ require clarification. The supplied record labels the problem open.

Sources

• Supplied local problem registry: permanent UnsolvedMath ID $30003634$, title Derived Pure Braid Groups in the Solvable Filtration, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied source citation: Mini-Workshop: Interactions between Low-dimensional Topology and Complex Algebraic Geometry (2018), Oberwolfach Reports, pp. 2987--3033, DOI 10.4171/owr/2017/50. Supplied report and literature-source link, labeled Oberwolfach Report 50/2017: https://doi.org/10.4171/owr/2017/50. The citation's 2018 date and the report's 2017 designation are retained without reconciliation.
• Supplied literature source labeled Otto: https://msp.org/agt/2014/14-5/agt-v14-n5-p05-s.pdf. The supplied record gives no fuller bibliographic details or pinpoint reference for the relevant claim.
• Literature assessment checked at 2026-08-21: supplied dated curation, not independently verified. The supplied status is open; the assessment says that only the base containment of the squared pure braid group in the zero-level filtration is established in the checked sources, while the stated higher-derived inclusion remains unresolved. The background characterizes this as dated literature triage derived from the supplied OWR curation and calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This is supplied editorial metadata, not independent verification of the formulation or its mathematical status.