Pure Braid Squares in the Symmetric Grope Filtration

Definitions and mathematical statement

For a positive integer $m$, let $P_m$ denote the pure braid group on $m$ labelled strands: its elements are isotopy classes of braids whose strands return to their respective starting positions, with multiplication given by stacking. The supplied record does not specify the intended range or quantification of $m$.

The rewritten source calls $(P_m)^2$ the square subgroup. Under the literal subgroup-of-squares interpretation, this means \[ (P_m)^2=\langle \beta^2\mid \beta\in P_m\rangle. \] This interpretation requires confirmation: the supplied original statement gives only the notation $(P_m)^2$, without defining it.

The record describes $\mathcal G_2^m$ as the second stage of the symmetric grope filtration. It does not supply the ambient group, the geometric definition of this filtration, its indexing convention, or the identification or map through which pure braids are compared with its elements. These missing specifications are necessary to make the containment precise.

With the intended source conventions still to be supplied, the reported question is \[ (P_m)^2\stackrel{?}{\subseteq}\mathcal G_2^m. \] Under the literal square-subgroup reading, this asks whether every finite product of squares of pure braids and their inverses belongs to the indicated second filtration stage, after the intended ambient identification. This is a conditional formulation of the supplied question, not a resolution of its missing conventions.

Short English statement

The question asks whether the square subgroup of the pure braid group lies in the second symmetric grope filtration stage. The meaning of the square notation, the filtration and ambient identification, and the quantification over $m$ require confirmation. The supplied record labels the problem open; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30003635$, exact source problem code catalogue entry, title "Pure Braid Squares in the Symmetric Grope Filtration." The original statement is $ (P_m)^2\subset G_2^m$, posed as a question. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Mini-Workshop: Interactions between Low-dimensional Topology and Complex Algebraic Geometry (2018), Oberwolfach Reports, pp. 2987--3033, DOI 10.4171/owr/2017/50. https://doi.org/10.4171/owr/2017/50. The literature-source label is "Oberwolfach Report 50/2017."
• Supplied literature source labelled "Otto": https://msp.org/agt/2014/14-5/agt-v14-n5-p05-s.pdf. The record supplies no further bibliographic details for this reference.
• Supplied literature assessment, checked 2026-08-21: containment in the second Whitney-tower stage is described as known, while the stronger containment in the second symmetric-grope subgroup is described as open. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment; both its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, and sentence structure were repaired. This is supplied editorial metadata, not independent mathematical verification of this draft.