Formal language theory and the geometry of 3-manifolds

Automatic groups were introduced in connection with geometric problems, in particular with the study of fundamental groups of 3-manifolds. In this article the class of automatic groups is extended to include the fundamental group of every compact 3-manifold which satisfies Thurston's geometrization conjecture. Toward this end, the class of asynchronously groups is introduced and studied, where is an arbitrary full abstract family of languages. For example may be the family of regular languagesReg, context-free languagesCF, or indexed languagesInd. The class consists of precisely those groups which are asynchronously automatic. It is proved that contains all of the above fundamental groups, but that does not. Indeed a virtually nilpotent group belongs to if and only if it is virtually abelian.

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