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2004 | 2 | 2 | 218-249
Tytuł artykułu

Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold

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EN
The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B 4,4,4ℍ3. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B 4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation of G is an irreducible component of the first homology when G admits certain symmetries. As a special case of the motivating argument the criterion is applied to principal congruence subgroups of B 4,4,4. The group theoretic computation shows that most of the, possibly nonprincipal, congruence subgroups are of positive first Betti number.
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
2
Strony
218-249
Opis fizyczny
Daty
wydano
2004-04-01
online
2004-04-01
Twórcy
Bibliografia
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  • [2] R.W. Carter: Finite groups of Lie type, Conjugacy classes and complex character, John Wiley and Sons, Chichester-New York-Brisbane-Tronto-Singapore, 1985.
  • [3] L.E. Dickson: Linear groups with an, exposition of the Galois field theory, Dover Publ., New York, 1958.
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  • [6] H.M. Hilden, M.T. Lozano and J.M. Montesinos “On the Borromean orbifolds: Geometry and arithmetics”, In: B. Apanasov, W.D. Neuman, A.W. Reid and L. Siebenmann (Eds.): Topology '90, de Gruyter, Berlin, 1992, pp. 133–167.
  • [7] H.M. Hilden, M.T. Lozano and J.M. Montesinos: “On the universal groups of the Borromean rings”, In: B. Apanasov, W.D. Neuman, A.W. Reid and L. Siebenmann (Eds.), Proceedings of the 1987 Siegen confernece on Differential Topology, Springer Verlag, Berlin, 1988, pp. 1–13.
  • [8] N. Jacobson: “Basic Algebra I, II”, 2nd Ed., Freeman, New York, 1985.
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  • [10] J.J. Millson: “On the first Betti number of a constant negatively curved manifold”, Jour. Ann. of Math., Vol. 104, (1976), pp. 235–247. http://dx.doi.org/10.2307/1971046
  • [11] J.G. Ratcliffe: Foundation of hyperbolic manifolds, Springer, New York, 1994.
  • [12] A.W. Reid: Arithmetic Kleinian groups and their Fuchsian subgroups, Thesis (PhD), University of Aberdeen University of Aberdeen, 1985.
  • [13] M. Suzuki: Group Theory I, II, Springer Verlag, New York, 1985.
  • [14] W.P. Thurston: “Three dimensional manifolds, Kleinian groups and hyperbolic geometry”, Jour. Bull. of AMS, Vol. 6, No. 3, (1982), pp. 357–381. http://dx.doi.org/10.1090/S0273-0979-1982-15003-0
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Bibliografia
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bwmeta1.element.doi-10_2478_BF02476541
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