Asphericity of groups defined by graphs

TL;DR

This paper introduces the concept of aspherical graphs and proves that such graphs define aspherical group presentations, generalizing existing theorems and providing new graphical conditions for asphericity.

Contribution

The paper defines aspherical graphs and proves their associated groups are aspherical, extending previous results and offering new conditions for asphericity in graphical group presentations.

Findings

Aspherical graphs define aspherical group presentations.

Generalization of Gruber's theorem on graphical $C(6)$-condition.

New graphical conditions for asphericity analogous to classical conditions.

Abstract

A graph $\Gamma$ labelled by a set $S$ defines a group $G(\Gamma)$ whose generators are the set of labels $S$ and whose relations are all words which can be read on closed paths of this graph. We introduce the notion of aspherical graph and prove that such a graph defines an aspherical group presentation. This result generalizes a theorem of Dominik Gruber on graphs satisfying graphical $C(6)$-condition and also allows to get new graphical conditions of asphericity analogous to some classical conditions.