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.
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References
[A]A. V. Aho,Indexed grammars—an extension of context-free grammars, J. Assoc. Comp. Mach.15 (1968), 647–671.
[AB]J. M. Alonso andM. R. Bridson,Semihyperbolic groups, Proc. London Math. Soc.70 (1993), 56–114.
[B1]M. R. Bridson,On the geometry of normal forms in discrete groups, Proc. London Math. Soc. (3)67 (1993), 596–616.
[B2]M. R. Bridson,Combings of semidirect products and 3-manifold groups, J. Geom. and Funct. Anal. (GAFA)3 (1993), 263–278.
[B3]M. R. Bridson,Regular combings, nonpositive curvature and the quasiconvexity of abelian subgroups, J. Pure and Appl. Algebra88 (1993), 23–35.
[Ba]G. Baumslag,Lecture Notes on Nilpotent Groups, Conf. Board of the Math. Sci. Regional Conf. Series in Math., vol. 2, Amer. Math. Soc., Providence, RI 1971.
[BGSS]G. Baumslag, S. Gersten, M. Shapiro andH. Short,Automatic groups and amalgams J. Pure and Appl. Algebra76 (1991), 229–316.
[BG]M. R. Bridson andR. H. Gilman,On bounded languages and the geometry of nilpotent groups, inCombinatorial and Geometric Group Theory (Edinburgh, 1993), LMS Lecture Notes (204) Cambr. Univ. Pr. 1995, 1–15.
[Br]N. Brady,The geometry of asynchronous automatic structures, Ph.D. Thesis, UC Berkeley, 1993.
[C]J. Cannon,The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata16 (1984), 123–148.
[CJ]A. Casson andD. Jungreis,Convergence groups and Seifert fibered 3-manifolds, Invent. Math.118 (1994), 441–456.
[E+]D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson andW. P. Thurston,Word processing in groups, Jones and Bartlett, Boston, MA, 1992.
[G]M. Gromov,Hyperbolic Groups, inEssays in group theory, S. M. Gersten (ed.), MSRI Publ. vol. 8, Springer Verlag, New York, 1987, 75–263.
[Ga]David Gabai,Convergence groups are Fuchsian groups, Annals of Mathematics136 (1992), 447–519.
[GS]S. Ginsburg andE. H. Spanier,Bounded Algol-like languages, Trans. A.M.S.113 (1964), 333–368.
[GSh]S. M. Gersten andH. B. Short,Rational subgroups of biautomatic groups, Ann. of Math. (2)134 (1991), 125–158.
[H]J. Hempel,3-manifolds, Annals of Math. Studies 86, Princ. Univ. Press, Princeton, NJ, 1976.
[Ha]Marshall Hall, Jr. The Theory of Groups, Macmillan, New York, 1959.
[HU]J. E. Hopcroft andJ. D. Ullman,Introduction to Automata Theory, Languages and Computation, Addison-Wesley, Reading, MA, 1979.
[JS]W. Jaco andP. Shalen,Seifert fibred spaces in 3-manifolds, Memoirs of the AMS 220, AMS, Providence, RI, 1979.
[J]K. Johannson,Homotopy equivalences of 3-manifolds with boundary, Springer Lecture Notes in Math.761, Springer Verlag, Berlin, 1979.
[K]H. Kneser,Geschlossen Flächen in dreidimensionalen Mannigfaltigkeiten, Jahr. der Deutscher math. Ver.38 (1929), 248–260.
[Lo]M. Lothaire,Combinatorics on Words, Addison-Wesley, Reading, MA, 1983.
[Mi]J. W. Milnor,A unique factorization theorem for 3-manifolds, Amer. J. Math.84 (1962), 1–7.
[Mo]J. Morgan,On Thurston's uniformization conjecture for three dimensional manifolds, inThe Smith conjecture, H. Bass and J. Morgan (eds.), Academic Press, Orlando, Fla., 1984.
[MS]D. E. Muller andP. E. Schupp,Groups, the theory of ends and context-free languages, J. Computer and System Sciences26 (1983), 295–310.
[S]G. P. Scott,The geometry of 3-manifolds, Bull. LMS15 (1983), 401–487.
[Sha]M. Shapiro,Automatic structure and graphs of groups, inTopology '90, Proceedings of the research semester in low-dimensional topology at Ohio State, de Gruyter Verlag, 1992.
[T1]W. P. Thurston,The geometry and topology of 3-manifolds, preprint, Princeton University, 1991.
[T2]W. P. Thurston,Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. AMS6 (1982), 357–381.
[W]J. A. Wolf,Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Diff. Geom.2 (1968), 421–446.
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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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Bridson, M.R., Gilman, R.H. Formal language theory and the geometry of 3-manifolds. Commentarii Mathematici Helvetici 71, 525–555 (1996). https://doi.org/10.1007/BF02566435
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DOI: https://doi.org/10.1007/BF02566435