Formal language theory and the geometry of 3-manifolds
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Commentarii Math.\ Helv., {\bf 71}, 1996, 525-555. (with Martin Bridson).


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 C(A) of
asynchronously A-combable groups is introduced and studied, where A is an
arbitrary full abstract family of languages.  For example A may be the
family of regular languages, Reg, context--free languages, CF, or indexed
languages, Ind.  The class C(Reg) consists of precisely those groups
which are asynchronously automatic.  It is proved that C(Ind) contains
all of the above fundamental groups, but that C(CF) does not.  Indeed a
virtually nilpotent group belongs to C(CF) if and only if it is virtually
abelian.