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 A(n), B-n = C-n, and affine type (A) over bar (n-1), (C) over bar (n-1). The Artin group A(A(n)) is isomorphic to the (n + 1)-strand braid group Bn+1, and the other three Artin groups are isomorphic to some subgroups of Bn+1. The inclusions between these subgroups yield monomorphisms A(B-n) -> A(A(n)), A((A) over bar (n-1)) -> A(B-n) and A((C) over bar (n-1)) -> A(B-n). There are another type of monomorphisms A(B-d) -> A(A(md-1)), A(B-d) -> A(B-md) and A(B-d) -> A(A(md)) which are induced by isomorphisms between Artin groups of type B and centralizers of periodic braids. In this paper, we show that the monomorphisms A(B-d) -> A(A(md-1)), A(B-d) -> A(B-md) and A(B-d) -> A(A(md)) induce injective functions on the set of conjugacy classes, and that none of the monomorphisms A(B-n) -> A(A(n)), A((A) over bar (n-1)) -> A(B-n) and A((C) over bar (n-1)) -> A(B-n) does so. (C) 2008 Elsevier Inc. All rights reserved.