Book contents
- Frontmatter
- Contents
- Foreword
- Participants
- On bounded languages and the geometry of nilpotent groups
- Finitely presented groups and the finite generation of exterior powers
- Semigroup presentations and minimal ideals
- Generalised trees and Λ-trees
- The mathematician who had little wisdom: a story and some mathematics
- Palindromic automorphisms of free groups
- A Freiheitssatz for certain one-relator amalgamated products
- Isoperimetric functions of groups and exotic cohomology
- Some embedding theorems and undecidability questions for groups
- Some results on bounded cohomology
- On perfect subgroups of one-relator groups
- Weight tests and hyperbolic groups
- A non-residually finite, relatively finitely presented group in the variety N2A
- Hierarchical decompositions, generalized Tate cohomology, and groups of type (FP)∞
- Tree-lattices and lattices in Lie groups
- Generalisations of Fibonacci numbers, groups and manifolds
- Knotted surfaces in the 4-sphere with no minimal Seifert manifolds
- The higher geometric invariants of modules over Noetherian group rings
- On calculation of width in free groups
- Hilbert modular groups and isoperimetric inequalities
- On systems of equations in free groups
- Cogrowth and essentiality in groups and algebras
- Regular geodesic languages for 2-step nilpotent groups
- Finding indivisible Nielsen paths for a train track map
- More on Burnside's problem
- Problem Session
On perfect subgroups of one-relator groups
Published online by Cambridge University Press: 05 April 2013
- Frontmatter
- Contents
- Foreword
- Participants
- On bounded languages and the geometry of nilpotent groups
- Finitely presented groups and the finite generation of exterior powers
- Semigroup presentations and minimal ideals
- Generalised trees and Λ-trees
- The mathematician who had little wisdom: a story and some mathematics
- Palindromic automorphisms of free groups
- A Freiheitssatz for certain one-relator amalgamated products
- Isoperimetric functions of groups and exotic cohomology
- Some embedding theorems and undecidability questions for groups
- Some results on bounded cohomology
- On perfect subgroups of one-relator groups
- Weight tests and hyperbolic groups
- A non-residually finite, relatively finitely presented group in the variety N2A
- Hierarchical decompositions, generalized Tate cohomology, and groups of type (FP)∞
- Tree-lattices and lattices in Lie groups
- Generalisations of Fibonacci numbers, groups and manifolds
- Knotted surfaces in the 4-sphere with no minimal Seifert manifolds
- The higher geometric invariants of modules over Noetherian group rings
- On calculation of width in free groups
- Hilbert modular groups and isoperimetric inequalities
- On systems of equations in free groups
- Cogrowth and essentiality in groups and algebras
- Regular geodesic languages for 2-step nilpotent groups
- Finding indivisible Nielsen paths for a train track map
- More on Burnside's problem
- Problem Session
Summary
If P is a group of operators on a group A, then we denote by dp(A) the minimal number of generators of A as a P-group.
Suppose G is a group and F/N is a presentation of G. Then F acts on N by conjugation and induces an action of G on Nab. This ℤG-module is called the relation module of the presentation F/N.
Definition. A presentation F/N of a group G is said to have a relation gap if dG(Nab) is strictly less than dF(N).
It is an open problem whether there exists a presentation that has a relation gap (see Harlander [H1], [H2] and Baik, Pride [B-P]). Such a presentation would be interesting not only to group theorists. In [D] Dyer shows that a presentation with a relation gap could be used to settle an open question concerning complexes dominated by a 2-complex (see also Wall [W] and Ratcliffe [R]).
In [H1] and [H2] the author studies groups that have cyclic relation modules. Such groups are quotients G/P, with G a one-relator group, say presented by 〈X ∣ r〉, and P a perfect normal subgroup G. Now if P is not of the form 〈w〉F / 〈r〉F, where 〈w〉F denotes the normal closure of the element w of the free group F on X, then G/P has a presentation with a relation gap.
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- Combinatorial and Geometric Group Theory, Edinburgh 1993 , pp. 164 - 173Publisher: Cambridge University PressPrint publication year: 1994