Mathematical Research Letters

Volume 20 (2013)

Number 1

A presentation for the pure Hilden group

Pages: 181 – 202



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


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.

Published 20 September 2013