Injectivity on the set of conjugacy classes of some monomorphisms between Artin groups

There are well-known monomorphisms between the Artin groups of finite type $\arA_n$, $\arB_n=\arC_n$ and affine type $\tilde \arA_{n-1}$, $\tilde\arC_{n-1}$. The Artin group $A(\arA_n)$ is isomorphic to the $(n+1)$-strand braid group $B_{n+1}$, and the other three Artin groups are isomorphic to some subgroups of $B_{n+1}$. The inclusions between these subgroups yield monomorphisms $A(\arB_n)\to A(\arA_n)$, $A(\tilde \arA_{n-1})\to A(\arB_n)$ and $A(\tilde \arC_{n-1})\to A(\arB_n)$. There are another type of monomorphisms $A(\arB_d)\to A(\arA_{md-1})$, $A(\arB_d)\to A(\arB_{md})$ and $A(\arB_d)\to A(\arA_{md})$ which are induced by isomorphisms between Artin groups of type $\arB$ and centralizers of periodic braids. In this paper, we show that the monomorphisms $A(\arB_d)\to A(\arA_{md-1})$, $A(\arB_d)\to A(\arB_{md})$ and $A(\arB_d)\to A(\arA_{md})$ induce injective functions on the set of conjugacy classes, and that none of the monomorphisms $A(\arB_n)\to A(\arA_n)$, $A(\tilde \arA_{n-1})\to A(\arB_n)$ and $A(\tilde \arC_{n-1})\to A(\arB_n)$ does so.

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