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

Masahito Toda

Open Mathematics (2004)

  • Volume: 2, Issue: 2, page 218-249
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Masahito Toda. "Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold." Open Mathematics 2.2 (2004): 218-249. <http://eudml.org/doc/268864>.

@article{MasahitoToda2004,
abstract = {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.},
author = {Masahito Toda},
journal = {Open Mathematics},
keywords = {hyperbolic geometry; 3-manifold; arithmetic lattice; finite groups of Lie type; MSC (2000); 57M12; 57M50; 57M60; 57S17; 20C05; 20C33; Betti number},
language = {eng},
number = {2},
pages = {218-249},
title = {Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold},
url = {http://eudml.org/doc/268864},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Masahito Toda
TI - Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold
JO - Open Mathematics
PY - 2004
VL - 2
IS - 2
SP - 218
EP - 249
AB - 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.
LA - eng
KW - hyperbolic geometry; 3-manifold; arithmetic lattice; finite groups of Lie type; MSC (2000); 57M12; 57M50; 57M60; 57S17; 20C05; 20C33; Betti number
UR - http://eudml.org/doc/268864
ER -

References

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  10. [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 Zbl0364.53020
  11. [11] J.G. Ratcliffe: Foundation of hyperbolic manifolds, Springer, New York, 1994. 
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