Presentations for Subsemigroups of Groups
Ph.D. thesis, University of St Andrews, St Andrews, 2005.

Abstract

This thesis studies subsemigroups of groups from three perspectives: automatic structures, ordinary semigroup presentations, and Malcev presentations. [A Malcev presentation is a presentation of a special type for a semigroup that can be embedded into a group. A group-embeddable semigroup is Malcev coherent if all of its finitely generated subsemigroups admit finite Malcev presentations.]

The theory of synchronous and asynchronous automatic structures for semigroups is expounded, particularly for group-embeddable semigroups. In particular, automatic semigroups embeddable into groups are shown to inherit many of the pleasant geometric properties of automatic groups. It is proved that group-embeddable automatic semigroups admit finite Malcev presentations, and such presentations can be found effectively. An algorithm is exhibited to test whether an automatic semigroup is free. Cancellativity of automatic semigroups is proved to be undecidable.

Study is made of several classes of groups: virtually free groups, groups that satisfy semigroup laws (in particular [virtually] nilpotent and [virtually] abelian groups), polycyclic groups, free products of groups, direct products of groups, and one-relator groups. For each of these classes, the question of Malcev coherence is considered, together with the problems of whether finitely generated subsemigroups are finitely presented or automatic. For subsemigroups of certain classes, the decidability of particular properties is considered. This study yields closure and containment results regarding the class of Malcev coherent groups.

The property of having a finite Malcev presentation is shown to be preserved under finite Rees index extensions and subsemigroups. Other concepts of index are also studied.