Mathematical Research Letters

Volume 20 (2013)

Number 1

A presentation for the pure Hilden group

Pages: 181 – 202

DOI: http://dx.doi.org/10.4310/MRL.2013.v20.n1.a14

Author

Stephen Tawn (Centre for Research in Mathematics, University of Western Sydney, Penrith, Australia)

Abstract

Consider the half ball, ${\mathbb{B}}^3_+$, containing $n$ unknotted and unlinked arcs $a_1, a_2, \ldots, a_n$ such that the boundary of each $a_i$ lies in the plane. The Hilden (or Wicket) group is the mapping class group of ${\mathbb{B}}^3_+$ fixing the arcs $a_1 \cup a_2 \cup \cdots \cup a_n$ setwise and fixing the half sphere ${\mathbb{S}}^2_+$ pointwise. This group can be considered as a subgroup of the braid group on $2n$ strands. The pure Hilden group is defined to be the intersection of the Hilden group and the pure braid group. In a previous paper, we computed a presentation for the Hilden group using an action of the group on a cellular complex. This paper uses the same action and complex to calculate a finite presentation for the pure Hilden group. The framed braid group acts on the pure Hilden group by conjugation and this action is used to reduce the number of cases.

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