Uniqueness of roots up to conjugacy for some affine and finite type Artin groups

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Let G be one of the Artin groups of finite type Bₙ=Cₙ, and affine type Aₙ₋₁ and Cₙ₋₁. In this paper, we show that if α and β are elements of G such that αᵏ=βᵏ for some nonzero integer k, then α and β are conjugate in G. For the Artin group of type Aₙ, this was recently proved by J. González-Meneses. In fact, we prove a stronger theorem, from which the above result follows easily by using descriptions of those Artin groups as subgroups of the braid group on n+1 strands. Let P be a subset of {1,...,n}. An n-braid is said to be P-pure if its induced permutation fixes each i∈ P, and P-straight if it is P-pure and it becomes trivial when we delete all the i-th strands for i∈ P. Exploiting the Nielsen-Thurston classification of braids, we show that if α and β are P-pure n-braids such that αᵏ=βᵏ for some nonzero integer k, then there exists a P-straight n-braid γ with β=γαγ⁻¹. Moreover, if 1∈ P, the conjugating element γ can be chosen to have the first strand algebraically unlinked with the other strands. Especially in case of P={1,...,n}, our result implies the uniqueness of root of pure braids, which was known by V. G. Bardakov and by D. Kim and D. Rolfsen.