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

TL;DR

This paper investigates how certain monomorphisms between Artin groups preserve conjugacy classes, showing some induce injective functions while others do not, thus clarifying their structural properties.

Contribution

It proves that specific monomorphisms between Artin groups induce injective functions on conjugacy classes, providing new insights into their algebraic structure.

Findings

Monomorphisms $A(ar{B}_d) o A(ar{A}_{md-1})$ induce injective functions on conjugacy classes.

Monomorphisms $A(ar{B}_d) o A(ar{B}_{md})$ induce injective functions on conjugacy classes.

Other monomorphisms like $A(ar{B}_n) o A(ar{A}_n)$ do not induce injective functions on conjugacy classes.

Abstract

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.