The structure of the group of conjugating automorphisms and the linear representation of the braid groups of some manifolds

V. G. Bardakov

paper

The structure of the group of conjugating automorphisms and the linear representation of the braid groups of some manifolds

arXiv:math/0301247

Abstract

In this paper we describe the structure of a group of conjugating automorphisms $C_{n}$ of free group and prove that this structure is similar to the structure of a braid group $B_{n}$ with $n > 1$ strings. We find the linear representation of group $C_{n}$ . Also we prove that the braid group $B_{n} (S^{2})$ of 2--sphere, mapping class group M(0,n) of the $n$ --punctured 2--sphere and the braid group $B_{3} (P^{2})$ of the projective plane are linear. Using result of J. Dyer, E. Formanek, E. Grossman and the faithful linear representation of Lawrence--Krammer of $B_{4}$ we construct faithful linear representation of the automorphism group $A u t (F_{2})$ .

24 pages

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