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

Abstract

From a group H and h e H, we define a representation ρ: B n → Aut(H* n ), where B n denotes the braid group on n strands, and H* n denotes the free product of n copies of H. We call p 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 × ρ B n , where ρ: B n → Aut(H* n ) is an Artin type representation. In particular, we show that H* n × ρ B n is a Garside group if H is a Garside group and h is a Garside element of H.