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

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

2002-12-10

arxiv.org/abs/math/0212138v1

Mathematics

Group Theory

Scientific paper

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 $H$.

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