Asphericity of Groups Defined by Graphs

Abstract

A graph Γ labeled by a set S defines a group G(Γ) whose set of generators is the set S of labels and whose relations are all words which can be read on closed paths of this graph. We introduce the notion of an aspherical graph and prove that such a graph defines an aspherical group presentation. This result generalizes a theorem of Dominik Gruber on graphs satisfying the graphical C(6)-condition and makes it possible to obtain new graphical conditions of asphericity similar to some classical conditions.