Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups

Abstract

From a group $H$ and a non-trivial element $h$ of $H$, we define a representation $\rho: B_n \to \Aut(G)$, where $B_n$ denotes the braid group on $n$ strands, and $G$ denotes the free product of $n$ copies of $H$. Such a representation shall be called the Artin type representation associated to the pair $(H,h)$. The goal of the present paper is to study different aspects of these representations. Firstly, we associate to each braid $\beta$ a group $\Gamma_{(H,h)} (\beta)$ and prove that the operator $\Gamma_{(H,h)}$ determines a group invariant of oriented links. We then give a topological construction of the Artin type representations and of the link invariant $\Gamma_{(H,h)}$, and we prove that the Artin type representations are faithful. The last part of the paper is dedicated to the study of some semidirect products $G \rtimes_\rho B_n$, where $\rho: B_n \to \Aut(G)$ is an Artin type representation. In particular, we show that $G \rtimes_\rho B_n$ is a Garside group if $H$ is a Garside group and $h$ is a Garside element of $H$.