Equality of the Andreadakis and Representation-Ring Filtrations

Definitions and mathematical statement

Let $F_n$ be the free group of rank $n$. Define its lower central series by \[ \gamma_1(F_n)=F_n,\qquad \gamma_{r+1}(F_n)=[F_n,\gamma_r(F_n)]\quad(r\geq1). \] Using the standard indexing of the Andreadakis--Johnson filtration, put \[ A_n(k)=\ker\!\left(\operatorname{Aut}(F_n)\longrightarrow\operatorname{Aut}\left(F_n/\gamma_{k+1}(F_n)\right)\right),\qquad k\geq1. \] Write $R_n$ for the object called the universal $\operatorname{SL}_2({\mathbb{C}})$-representation ring of $F_n$ in the supplied record, and write $J\subseteq R_n$ for its augmentation ideal. The record does not supply a construction of $R_n$ or its augmentation. Subject to the intended construction and the stated action of $\operatorname{Aut}(F_n)$ preserving $J$, define \[ D_n(k)=\ker\!\left(\operatorname{Aut}(F_n)\longrightarrow\operatorname{Aut}\left(J/J^{k+1}\right)\right). \] Here $J^{k+1}$ is the ordinary ideal power, and the target denotes automorphisms of the quotient with the structure preserved by the intended action. Equivalently, \[ D_n(k)=\{\varphi\in\operatorname{Aut}(F_n):\varphi\cdot u-u\in J^{k+1}\text{ for every }u\in J\}. \] Conjecture as supplied. For the intended rank $n$, \[ A_n(k)=D_n(k)\qquad\text{for every integer }k\geq1. \] The supplied record does not specify the allowed ranks or whether the assertion is universally quantified over $n$. It also leaves the precise representation-ring construction and augmentation unresolved; consequently, the definition of $D_n(k)$ and the conjecture remain conditional on those choices. The displayed definition of $A_n(k)$ makes the standard indexing convention explicit.

Short English statement

The conjecture asks whether an automorphism of a free group acts trivially on its nilpotent quotient of class $k$ exactly when it acts trivially on $J/J^{k+1}$, for every $k\geq1$. The intended representation ring, its augmentation, and the range of ranks require clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Local registry: supplied complete source record, permanent UnsolvedMath ID $30003703$, source problem code catalogue entry, title Equality of the Andreadakis and Representation-Ring Filtrations, in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Topology of Arrangements and Representation Stability (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/2, pp. 43--123. Supplied source URL and literature-sources link, labeled "Oberwolfach report": https://doi.org/10.4171/owr/2018/2.
• Attribution: no proposer is named in the supplied record. Its proposed year and workshop-year tag are 2018; the report title citation gives 2019. These supplied dates are retained without resolving their relationship.
• Supplied literature assessment, dated exactly 2026-08-21: the record labels the problem open, describes the equality as a conjecture, and reports that no later proof or counterexample was located. This is supplied dated curation, not an independently verified literature assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry gives the statement status as corrected_verified and reports that the official report was checked and that the reconstruction restored both filtrations and the representation-ring augmentation ideal. These are supplied verification claims; no independent source visit or mathematical verification is asserted here.