Level-two congruence subgroup conjecture for non-spherical small Coxeter groups

Let Γ\Gamma be a non-spherical small Coxeter graph, with associated Artin group A[Γ]A[\Gamma], Coxeter group W[Γ]W[\Gamma], and level-two congruence subgroup A[Γ][2]A[\Gamma][2]. Write W[Γ][2]W[\Gamma][2] for the level-two congruence subgroup of W[Γ]W[\Gamma]. Level-two congruence subgroup conjecture.

A[Γ][2]=ker(A[Γ]W[Γ]/W[Γ][2]).A[\Gamma][2]=\ker\bigl(A[\Gamma]\to W[\Gamma]/W[\Gamma][2]\bigr).

The preceding argument establishes the inclusion of the right-hand kernel in A[Γ][2]A[\Gamma][2]; the conjecture asserts equality for every non-spherical small Coxeter graph. The parser provides no evidence of a resolution, so the conjecture remains open.

Sources & referencesView supporting material

Primary source

Pravin Kumar, “Congruence subgroups of small Artin and Coxeter groups”, arXiv:2510.23002 (2025).

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