GRAPH GROUPS ARE BIAUTOMATIC

Graph groups admit a (finite) presentation in which each relation is of the form xy = yx for generators x and y. While the two extreme cases of graph groups, free groups and free abelian groups, have been previously shown to be bicombable (in fact, biautomatic), neither of the normal forms typically used for the combings generalize successfully to arbitrary graph groups. The normal forms presented here which do yield results for arbitrary graph groups utilize the concept of a ''commuting clique'' of generators, and when these normal forms are applied to free abelian groups, they differ from the ''usual'' normal forms. As the set of normal forms is a regular language over the free monoid on the set of generators and their formal inverses, it follows that graph groups are biautomatic.