A presentation for the pure Hilden group

2009/02/27 by Stephen Tawn, Tawn, Stephen
Mathematics · #20F05 #20F36 #57M07 #57M60 #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #math.GR #math.GT #msc:20F05 #msc:20F36 #msc:57M07 #msc:57M60

paper · pdf · doi:10.48550/arxiv.0902.4840

26 pages

arxiv created 2009/02/27 · openalex publication_date 2009/02/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the unit ball, B = D × [0,1], containing n unknotted arcs a₁, a₂, ..., a_n such that the boundary of each a_i lies in D × \0\. The Hilden (or Wicket) group is the mapping class group of B fixing the arcs a₁ ∪ a₂ ∪ ... ∪ a_n setwise and fixing D × \1\ pointwise. This group can be considered as a subgroup of the braid group. 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 presentaion 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.

Citations

• Configuration spaces of rings and wickets

Related

• The Hilden Double Coset Problem in Braid Groups
• Hilden Braid Groups
• The mixed Hilden braid group and the plat equivalence in handlebodies
• A presentation for the planar pure braid group
• Presentations of the braid group of the complex reflection group G(d,d,n)