Automatic Structures and Boundaries for Graphs of Groups

We study the synchronous and asynchronous automatic structures on the fundamental group of a graph of groups in which each edge group is finite. Up to a natural equivalence relation, the set of biautomatic structures on such a graph product bijects to the product of the sets of biautomatic structures on the vertex groups. The set of automatic structures is much richer. Indeed, it is dense in the infinite product of the sets of automatic structures of all conjugates of the vertex groups. We classify these structures by a class of labelled graphs which “mimic” the underlying graph of the graph of groups. Analogous statements hold for asynchronous automatic structures. We also discuss the boundaries of these structures.