Volume 10, issue 4 (2006)

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Three-manifolds, virtual homology, and group determinants

Daryl Cooper and Genevieve S Walsh

Geometry & Topology 10 (2006) 2247–2269

arXiv: math.GT/0603152

Abstract

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.

Keywords
Group determinant, Dehn-filling, virtually Haken
Mathematical Subject Classification 2000
Primary: 57M10
Secondary: 57M25, 20C10
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Publication
Received: 8 March 2006
Accepted: 24 October 2006
Published: 29 November 2006
Proposed: David Gabai
Seconded: Cameron Gordon, Joan Birman
Authors
Daryl Cooper
Math Department
UCSB
Santa Barbara, CA 93106
USA
Genevieve S Walsh
Department of Math
Tufts University
Medford, MA 02155
USA
and
Département de Mathématiques
UQAM
Montréal, QC H3C 3J7
Canada