Embedding Independence of Congruence Subgroups

Definitions and mathematical statement

Let $N\geq 1$ be an integer, let $G$ be a finitely generated group, and let $\iota:G\hookrightarrow\operatorname{SL}_N({\mathbb{Z}})$ be an injective group homomorphism. For each integer $m\geq 1$, define the principal congruence subgroup \[ \Gamma_N(m)=\ker\!\left(\operatorname{SL}_N({\mathbb{Z}})\longrightarrow\operatorname{SL}_N({\mathbb{Z}}/m{\mathbb{Z}})\right), \] where the map is reduction of matrix entries modulo $m$, and $\Gamma_N(1)=\operatorname{SL}_N({\mathbb{Z}})$. Use the standard convention that an ambient congruence subgroup is a subgroup $C\leq\operatorname{SL}_N({\mathbb{Z}})$ containing $\Gamma_N(m)$ for some $m\geq 1$. Transporting intersections back to the abstract group $G$, define \[ \mathcal C_{\iota}=\left\{\iota^{-1}(C): C\leq\operatorname{SL}_N({\mathbb{Z}}),\ \Gamma_N(m)\leq C\text{ for some }m\geq 1\right\}. \] Equivalently, with $K_{\iota}(m)=\iota^{-1}(\Gamma_N(m))$, this family consists of the subgroups $H\leq G$ containing $K_{\iota}(m)$ for at least one $m\geq 1$.

Question. For every integer $N\geq 1$, every finitely generated group $G$, and every pair of embeddings $\iota_1,\iota_2:G\hookrightarrow\operatorname{SL}_N({\mathbb{Z}})$, must \[ \mathcal C_{\iota_1}=\mathcal C_{\iota_2} \] hold? Equivalently, can the family of congruence subgroups of the abstract group $G$ depend on its chosen embedding?

Unresolved qualifications. The supplied wording does not explicitly say whether the target dimension must remain fixed when comparing embeddings. The displayed question compares embeddings into the same $\operatorname{SL}_N({\mathbb{Z}})$; allowing embeddings into $\operatorname{SL}_{N_1}({\mathbb{Z}})$ and $\operatorname{SL}_{N_2}({\mathbb{Z}})$ gives the corresponding broader formulation, whose intended inclusion requires source review. The definition above uses all ambient congruence subgroups, rather than only principal congruence subgroups; the supplied record does not explicitly spell out this convention.

Short English statement

A chosen integral matrix embedding gives congruence subgroups of $G$ by pulling back ambient subgroups containing a principal congruence subgroup. The question asks whether changing the embedding changes this family. Whether comparison across different matrix dimensions is intended remains unspecified.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $30003786$, source problem code catalogue entry, title Embedding Independence of Congruence Subgroups, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied source citation: Mini-Workshop: Superexpanders and Their Coarse Geometry (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/19, pp. 1117--1160. https://doi.org/10.4171/owr/2018/19. The registry gives the proposed year as 2018; the relationship between that year and the citation's 2019 label has not been verified.
• Supplied literature source, labelled "workshop report": https://ems.press/content/serial-article-files/46739.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, states that the report explicitly poses embedding dependence as open, and reports that no definitive resolution for this formulation was found by its cutoff. This is supplied dated curation, not an independently verified assessment. The background expressly requests 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 omitted definitions were repaired. These are supplied editorial assertions, not mathematical verification of this draft.