Geodesic
Three-manifolds, virtual homology, and group determinants
Cooper, Daryl1 ; Walsh, Genevieve S2
1 Math Department, UCSB, Santa Barbara, CA 93106, USA
2 Department of Math, Tufts University, Medford, MA 02155, USA, and, Département de Mathématiques, UQAM, Montréal, QC H3C 3J7, Canada
Geometry & topology, Tome 10 (2006) no. 4, pp. 2247-2269.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

We apply representation theory to study the homology of equivariant Dehn-fillings of a given finite, regular cover of a compact 3–manifold with boundary a torus. This yields a polynomial which gives the rank of the part of the homology carried by the solid tori used for Dehn-filling. The polynomial is a symmetrized form of the group determinant studied by Frobenius and Dedekind. As a corollary every such hyperbolic 3–manifold has infinitely many virtually Haken Dehn-fillings.

DOI : 10.2140/gt.2006.10.2247
Keywords: Group determinant, Dehn-filling, virtually Haken

Cooper, Daryl 1 ; Walsh, Genevieve S 2

1 Math Department, UCSB, Santa Barbara, CA 93106, USA
2 Department of Math, Tufts University, Medford, MA 02155, USA, and, Département de Mathématiques, UQAM, Montréal, QC H3C 3J7, Canada 
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Cooper, Daryl; Walsh, Genevieve S. Three-manifolds, virtual homology, and group determinants. Geometry & topology, Tome 10 (2006) no. 4, pp. 2247-2269. doi : 10.2140/gt.2006.10.2247. http://geodesic.mathdoc.fr/articles/10.2140/gt.2006.10.2247/

[1] M Burrow, Representation theory of finite groups, Academic Press (1965)

[2] D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5, Mathematical Society of Japan (2000)

[3] D Cooper, D D Long, Virtually Haken Dehn-filling, J. Differential Geom. 52 (1999) 173

[4] D Cooper, G S Walsh, Virtually Haken fillings and semi-bundles, Geom. Topol. 10 (2006) 2237

[5] N M Dunfield, W P Thurston, The virtual Haken conjecture: experiments and examples, Geom. Topol. 7 (2003) 399

[6] K W Johnson, The Dedekind–Frobenius group determinant: new life in an old problem, from: "Groups St. Andrews 1997 in Bath, II", London Math. Soc. Lecture Note Ser. 261, Cambridge Univ. Press (1999) 417

[7] T Y Lam, Representations of finite groups: a hundred years I, Notices Amer. Math. Soc. 45 (1998) 361

[8] D D Long, A W Reid, Simple quotients of hyperbolic 3–manifold groups, Proc. Amer. Math. Soc. 126 (1998) 877

[9] R Riley, Parabolic representations of knot groups I, Proc. London Math. Soc. $(3)$ 24 (1972) 217

[10] J A Sjogren, Connectivity and spectrum in a graph with a regular automorphism group of odd order, Internat. J. Algebra Comput. 4 (1994) 529

[11] J A Wolf, Spaces of constant curvature, Publish or Perish (1974)

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