Three-manifolds, virtual homology, and group determinants

Cooper, Daryl; Walsh, Genevieve S

doi:10.2140/gt.2006.10.2247

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Three-manifolds, virtual homology, and group determinants

Daryl Cooper and Genevieve S Walsh

Geometry & Topology 10 (2006) 2247–2269

DOI: 10.2140/gt.2006.10.2247

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

Volume 10, issue 4 (2006)