Presentation of the kernel of the transpositional action

Find a presentation for the kernel of the natural homomorphism from the Coxeter group C(Γ), associated with a connected simple graph Γ, to the symmetric group S_n acting on the n vertices of Γ.

Background

For a connected simple graph Γ with n vertices, the Coxeter group C(Γ) is generated by the edges of Γ and acts on the vertex set by sending each edge to the corresponding transposition. The authors describe the point stabilizers C(Γ)o and obtain the short exact sequence 1 → ⋂o∈V(Γ) C(Γ)o → C(Γ) → S_n → 1. The subgroup at the left is precisely the kernel of the transpositional action.

The paper develops presentations for the individual point stabilizers and notes that the corresponding kernel would be useful for computing fundamental groups of Galois covers. A related kernel for the quotient group CY(Γ) is known, but the presentation of the kernel for C(Γ) itself is left unresolved.

References

Question 6.10. Let Γ be a connected simple graph with n = |V (Γ)|. Find a presentation for the kernel of C(Γ)→Sn. A concise description of the kernel will be quite helpful for the computation of fundamental groups of Galois covers, see [1]. Note that the kernel of CY(Γ)→Sn was fully described in [18].

— Stabilizers in Coxeter groups with a transpositional action  (2609.01309 - Cohen et al., 1 Sep 2026) in Question 6.10, Section 6, page 19