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We consider groups defined by non-empty balanced presentations with theproperty that each relator is of the form R(x,y), where x and y are distinctgenerators and R(.,.) is determined by some fixed cyclically reduced wordR(a,b) that involves both a and b. To every such presentation we associate adirected graph whose vertices correspond to the generators and whose arcscorrespond to the relators. Under the hypothesis that the girth of theunderlying undirected graph is at least 4, we show that the resulting groupsare non-trivial and cannot be finite of rank 3 or higher. Without thehypothesis on the girth it is well known that both the trivial group and finitegroups of rank 3 can arise.

A class of digraph groups defined by balanced presentations