Isomorphism between the extended Artin group and the extended Seidel–Thomas braid group

Establish that the surjective homomorphism ιX: GA → ST(DX_A) defined by mapping each generator σ_v (for v in the vertex set V_A of the Coxeter–Dynkin diagram Γ_A) to the spherical twist T^X_{E_v} in the X-Calabi–Yau triangulated category DX_A and mapping τ to the spherical twist T_S is an isomorphism, thereby proving GA ≅ ST(DX_A) for the tuple A = (a1,a2,a3) with χ_A > 0.

Background

Section 5 constructs the extended Artin group GA associated to the Coxeter–Dynkin diagram Γ_A (Definition 5.4) and the extended Seidel–Thomas braid group ST(DX_A) of the X-Calabi–Yau triangulated category DX_A obtained as the X-Calabi–Yau completion of the octopus algebra CT_A (Definition 5.13).

Theorem 5.15 proves that the map ιX: GA → ST(DX_A) sending σv to the spherical twist T^X{E_v} and τ to T_S is a surjective homomorphism. Conjecture 5.16 asserts that this surjection is actually an isomorphism. The conjecture is known to hold for A = (1,p,q) and positive integers N ≥ 2, as referenced in [IUU, Qiu2, Wan].