Cyclic Presentations of the Trivial Group

Irreducible Cyclic Presentations of the Trivial Group

Balanced presentations of the trivial group on two generators and the Andrews-Curtis conjecture

The Andrews-Curtis conjecture states that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of the elementary Nielsen transformations and conjugations. In this paper we describe all balanced presentations of the trivial group on two generators and with the total length of relators ≤ 12. We show that all these presentations satisfy the Andrews-Curt...

The trivial group made

Coset enumeration, based on the methods described by Todd & Coxeter, is one of the basic tools for investigating nitely presented groups. The process is not well understood, and various pathological presentations of, for example, the trivial group have been suggested as challenge problems. Here we consider one such presentation. We show that it is much easier than it rst appears, albeit at the ...

On Rourke’s Extension of Group Presentations and a Cyclic Version of the Andrews–curtis Conjecture

In 1979, Rourke proposed to extend the set of cyclically reduced defining words of a group presentation P by using operations of cyclic permutation, inversion and taking double products. He proved that iterations of these operations yield all cyclically reduced words of the normal closure of defining words of P if the group, defined by the presentation P, is trivial. We generalize this result b...

Some Challenging Group Presentations

We study some challenging presentations which arise as groups of deeciency zero. In four cases we settle niteness: we show that two presentations are for nite groups while two are for innnite groups. Thus we answer three explicit questions in the literature and we provide the rst published deeciency zero presentation for a group with soluble length seven. The tools we use are coset enumeration ...