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

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

2003-01-22

arxiv.org/abs/math/0301247v2

Mathematics

Group Theory

24 pages

Scientific paper

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 $Aut(F_2)$.

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