Profinite Congruence Subgroup Problem for Mapping Class Groups

Definitions and mathematical statement

For a group $K$, its outer automorphism group is $\operatorname{Out}(K)=\operatorname{Aut}(K)/\operatorname{Inn}(K)$, where $\operatorname{Inn}(K)$ is the subgroup of automorphisms given by conjugation by elements of $K$.

The supplied problem asks whether the universal pro-$P$ outer monodromy representation $\rho^P_{g,r}$ is injective. The record does not define this representation, its domain or codomain, the meaning and permitted choices of $P$, or the admissible parameters $g,r$. Consequently, a fully specified mathematical statement cannot be recovered from the supplied text alone.

To express the requested property without choosing these missing data, let $D$ and $T$ stand for the intended domain and codomain groups of the named representation. Once those groups and the homomorphism \[ \rho^P_{g,r}\colon D\longrightarrow T \] have been specified, the question is whether \[ \forall x,y\in D,\qquad \rho^P_{g,r}(x)=\rho^P_{g,r}(y)\ \Longrightarrow\ x=y, \] or equivalently whether $\ker(\rho^P_{g,r})=\{1_D\}$. These placeholders only express injectivity; they do not identify the intended monodromy representation or supply the missing quantifiers over $P,g,r$.

The supplied dated assessment describes the problem as partially solved and reports injectivity for genus at most two, while describing the general congruence subgroup problem for mapping class groups as unresolved. This is supplied curation, not an independently verified theorem or status assessment. The record does not provide the hypotheses needed to state the reported low-genus results precisely, or establish the exact relationship between the named representation and the congruence subgroup formulation.

Short English statement

Does the named universal pro-$P$ outer monodromy representation distinguish every pair of elements of its domain? Its construction, parameter ranges, and hypotheses are missing from the supplied excerpt, so the displayed injectivity condition requires further source clarification. The supplied partially solved label and low-genus assessment remain unverified dated curation.

Sources

• Local UnsolvedMath registry: permanent ID $30005635$, source problem code catalogue entry, title "Profinite Congruence Subgroup Problem for Mapping Class Groups." The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is identified.
• Supplied source citation: MFO--RIMS Tandem Workshop: Arithmetic Homotopy and Galois Theory (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/42, pp. 2377--2488. https://doi.org/10.4171/owr/2023/42. This same reference is supplied under the literature-source label "OWR report with Asada--Boggi results."
• Supplied literature source: "Boggi, congruence topologies." https://arxiv.org/abs/1804.06322. No more detailed bibliographic information is supplied.
• Literature checked at: 2026-08-22. The supplied dated curation labels the problem partially solved, reports injectivity for genus at most two, and states that the general mapping class group congruence subgroup problem remains unresolved. The background explicitly requests independent verification of time-sensitive status and cited claims; none has been performed here.
• The registry labels the statement corrected_verified and reports comparison with the official Oberwolfach report. That verification claim is supplied metadata and has not been independently checked for this draft.