Gerald Williams (University of Essex)

Abstract: A cyclic presentation of a group is a group presentation with an equal number of generators and defining relations that admits a cyclic symmetry; the group it defines is called a cyclically presented group. Prominent examples include Fibonacci groups and Sieradski groups. Many 3-dimensional manifolds, such as all Dunwoody manifolds and the Fibonacci manifolds, have cyclically presented fundamental group. Frequently a stronger property holds, namely that the spine of the manifold is the presentation complex of a cyclic presentation of a group.
 

The focus of much prior research has concerned the construction of such manifolds. In this talk I will discuss a related problem that arises from a combinatorial group theory perspective. Namely, given a class of cyclic presentations, classify the presentations in that class that define fundamental groups of 3-manifolds, and classify the presentations whose presentation complexes are spines of 3-manifolds. I will discuss algebraic and topological tools that can be used to approach this problem, and present results concerning various generalisations of the Fibonacci groups. Aspects of this talk will incorporate joint work with Ihechukwu Chinyere, Vanni Noferini, and Jim Howie.