Serre’s conjectural classification of S-arithmetic groups with the congruence subgroup property

Classify S-arithmetic groups satisfying the congruence subgroup property in terms of their S-rank, thereby resolving the general statement of Serre’s conjecture.

Background

The paper discusses co-commensurability between the profinite completion and the congruence completion of an S-arithmetic group satisfying the congruence subgroup property. In this context, it refers to Serre’s conjecture, which proposes a classification of S-arithmetic groups with the congruence subgroup property according to their S-rank.

The authors explicitly note that the general form of this classification remains unresolved, although numerous important cases have already been established in the cited literature. This is an external open conjecture mentioned in the paper rather than a question addressed by the paper’s main results.

References

A well-known conjecture by Serre seeks to classify $S$-arithmetic groups satisfying the congruence subgroup property in terms of their $S$-rank . While the general statement is still open, many important cases have been proved .

— On commensurating graphs of pro-$\mathcal{C}$ groups  (2609.01234 - Cusinato, 1 Sep 2026) in Section 3, example following Corollary 3.?? (discussion of S-arithmetic groups and profinite completions)

Moreover, the congruence subgroup problem is open for second-type lattices and known to fail for first-type lattices.

— Complex hyperbolic surfaces whose closed geodesics are all simple  (2610.01806 - Douba et al., 1 Oct 2026) in Section 1, Introduction