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Linear representations of the group of conjugating automorphisms and the braid groups of some manifolds

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Abstract

We extend the Burau representation to the group C n of conjugating automorphisms. We extend the Lawrence—Krammer faithful linear representation of the braid group B3 to C3, and for n ≥ 4 we extend this representation under certain restrictions on the parameters of the representation. We determine that the spherical braid group B n (S2) and the mapping class group M(0, n) of an n-punctured sphere are linear for all n ≥ 2. The automorphism group Aut(F n ) is not linear for n ≥ 3, and the group Aut(F2) is linear iff so is the braid group B4. Using the Lawrence—Krammer representation, we construct a faithful linear representation of Aut(F2).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    V. G. Bardakov

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  1. V. G. Bardakov
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Original Russian Text Copyright © 2005 Bardakov V. G.

The author was supported by the Russian Foundation for Basic Research (Grant 02-01-01118).

Translated from Sibirski \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath} \) Matematicheski \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath} \) Zhurnal, Vol. 46, No. 1, pp. 17–31, January–February, 2005.

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Bardakov, V.G. Linear representations of the group of conjugating automorphisms and the braid groups of some manifolds. Sib Math J 46, 13–23 (2005). https://doi.org/10.1007/s11202-005-0002-5

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  • DOI: https://doi.org/10.1007/s11202-005-0002-5

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