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

Open Mathematics (2004)

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

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topMasahito 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 -

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