From a group H and h ∈ H,
we define a representation ρ : Bn→Aut(H∗n), where Bn denotes the braid group on
n strands, and H∗n denotes the free product of n copies of H. We call ρ the Artin
type representation associated to the pair (H,h). Here we study various aspects of
such representations.
Firstly, we associate to each braid β a group Γ(H,h)(β) and prove that the
operator Γ(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 Γ(H,h), and we prove that the Artin type representations are faithful
if and only if h is nontrivial. The last part of the paper is devoted to the
study of some semidirect products H∗n⋊ρBn, where ρ : Bn→Aut(H∗n)
is an Artin type representation. In particular, we show that H∗n⋊ρBn
is a Garside group if H is a Garside group and h is a Garside element of
H.