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Dedicated to the memory of Peter Stefan
Dedicated t o J o an Birman with gratitude and respect. Abstract. We prove the following result and show a connection with geometric intersection-number functions of curves on punctured discs and algebraic intersection functions for curves on surfaces: Let R be an associative ring with identity a n d x r 2 Z(R), where Z(R) i s the centre of R. Deene… (More)
We show that for n ≥ 3 there is an action of the braid group B n on the deter-minantal ideals of a certain n × n symmetric matrix with algebraically independent entries off the diagonal and 2s on the diagonal. We show how this action gives rise to an action of B n on certain compact subspaces of some Euclidean spaces of dimension ; n 2. These subspaces are… (More)
We construct representations for braid groups B n via actions of B n on a deter-minantal ring, thus mirroring the setting of the classical representation theory for GL n. The representations that we construct fix a certain unitary form. §1. Introduction Let C be a commutative ring with identity. In this paper we attempt to do for the braid groups B n what… (More)
Let the braid group B n act as (isotopy classes of) diffeomorphisms of an n-punctured disc D n. Then there is an action of B n on a polynomial algebra R = C [a 1 ,. .. , a N ] and a way of representing simple closed curves on D n as elements of R. Fix k ∈ 2N. Using this approach we show that the image in Aut(R) of each Dehn twist τ about a simple closed γ… (More)
We prove that the braid groups B n and their commutator subgroups B 0 n have an innnite number of distinct irreducible non-abelian complex representations in some xed dimensions. As a consequence we prove the same result for Aut(F 2).