9 research outputs found
Ribbon 2-knot groups of Coxeter type
Wirtinger presentations of deficiency 1 appear in the context of knots, long
virtual knots, and ribbon 2-knots. They are encoded by (word) labeled oriented
trees and, for that reason, are also called LOT presentations. These
presentations are a well known and important testing ground for the validity
(or failure) of Whitehead's asphericity conjecture. In this paper we define
LOTs of Coxeter type and show that for every given there exists a (prime)
LOT of Coxeter type with group of rank . We also show that label separated
Coxeter LOTs are aspherical
Minimal Seifert manifolds for higher ribbon knots
We show that a group presented by a labelled oriented tree presentation in
which the tree has diameter at most three is an HNN extension of a finitely
presented group. From results of Silver, it then follows that the corresponding
higher dimensional ribbon knots admit minimal Seifert manifolds.Comment: 33 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTMon1/paper12.abs.htm
Local indicability of groups with homology circle presentations
We investigate conditions that guarantee local indicability of groups that
admit presentations with the homology of a circle, generalizing a result of J.
Howie for two-relator presentations. We apply our results to investigate local
indicability of LOT groups and some classes of non-cycle-free Adian
presentations, extending previous results in that direction by J. Howie and D.
Wise.Comment: 15 page
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On the Logical Independence of (DR) and (CLA)
In 1941, J.H.C. Whitehead posed the question of whether asphericity is a hereditary property for 2-dimensional CW complexes. This question remains unanswered, but has led to the development of several algebraic and topological properties that are sufficient (but not necessary) for the asphericity of presentation 2-complexes. While many of the logical relationships between these flavors of asphericity are known, there remain a few to be answered. In particular, it has long been known that Cohen-Lyndon aspherical (CLA) complexes are not necessarily diagrammatically reducible (DR), but the existence of a (DR) complex which is not (CLA) remains open. We resolve this by showing that the presentation 2-complex associated to the presentation
is (DR) but not (CLA). In fact, we prove that if a nontrivial group G occurs as the fundamental group of a (DR) 2-complex, then there is a (DR) 2-complex with fundamental group that is not (CLA)
w-Cycles in Surface Groups
For w an element in the fundamental group of a closed, orientable, hyperbolic surface Ω which is not a proper power, and Σ a surface immersing in Ω, we show that the number of distinct lifts of w to Σ is bounded above by -χ(Σ). In special cases which can be characterised by interdependencies of the lifts of w, we find a stronger bound, whereby the total degree of covering from curves in Σ representing the lifts to the curve representing w is also bounded above by -χ(Σ). This is achieved by a method we introduce for decomposing surfaces into pieces that behave similarly to graphs, and using them to estimate Euler characteristics using a stacking argument of the kind introduced by Louder and Wilton. We then consider some consequences of these bounds for quotients of orientable surface groups by a single element. We demonstrate ways in which these groups behave analogously to one-relator groups; in particular, the ones with torsion are coherent (i.e. all finitely-generated subgroups have finite presentations), and those without torsion possess the related property of non-positive immersions as introduced by Wise