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Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups. (English) Zbl 1147.20033

Summary: From a group \(H\) and \(h\in H\), we define a representation \(\rho\colon B_n\to\operatorname{Aut}(H^{*n})\), where \(B_n\) denotes the braid group on \(n\) strands, and \(H^{*n}\) denotes the free product of \(n\) copies of \(H\). We call \(\rho\) the Artin type representation associated to the pair \((H,h)\). Here we study various aspects of such 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 if and only if \(h\) is nontrivial. The last part of the paper is devoted to the study of some semidirect products \(H^{*n}\rtimes_\rho B_n\), where \(\rho\colon B_n\to\operatorname{Aut}(H^{*n})\) is an Artin type representation. In particular, we show that \(H^{*n}\rtimes_\rho B_n\) is a Garside group if \(H\) is a Garside group and \(h\) is a Garside element of \(H\).

MSC:

20F36 Braid groups; Artin groups
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
20C15 Ordinary representations and characters
20E36 Automorphisms of infinite groups
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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