Formal language theory and the geometry of 3-manifolds |
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Authors: | Martin R Bridson Robert H Gilman |
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Institution: | (1) Department of Mathematics, Princeton University, 08544 Princeton, NJ;(2) Present address: Mathematical Institute, 24-29 St. Giles, OX1 3LB Oxford;(3) Department of Mathematics, Stevens Institute of Technology, 07030 Hoboken, NJ |
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Abstract: | 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.
The first author was partially supported by NSF grant DMS-9203500 and FNRS (Suisse). He also wishes to thank the University
of Geneva for its hospitality while this paper was being written. The second author thanks the Institute for Advanced Study
for its hospitality while this paper was being written. |
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Keywords: | |
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