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Graph groups are biautomatic. (English) Zbl 0812.20018
Author’s summary: 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 in this paper 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.

MSC:
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M07 Topological methods in group theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20F28 Automorphism groups of groups
20E05 Free nonabelian groups
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