A semigroup theoretic approach to Whitehead's asphericity question

Abstract

The Whitehead asphericity problem, regarded as a problem of combinatorial group theory, asks whether any subpresentation of an aspherical group presentation is also aspherical. This is a long standing open problem which has attracted a lot of attention. Related to it, throughout the years there have been given several useful characterizations of asphericity which are either combinatorial or topological in nature. The aim of this paper is two fold. First, it brings in methods from semigroup theory to give a new combinatorial characterization of asphericity in terms of what we define here to be the weak dominion of a submonoid of a monoid, and uses this to give a sufficient and necessary condition under which a subpresentation of an aspherical group presentation is aspherical.