Vol. 229, No. 1, 2007

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ISSN: 0030-8730
Handle additions producing essential surfaces

Ruifeng Qiu and Shicheng Wang

Vol. 229 (2007), No. 1, 233–255

We construct a small, hyperbolic 3-manifold M with the property that, for any integer g 2, there are infinitely many separating slopes r in ∂M such that the 3-manifold M(r) obtained by attaching a 2-handle to M along r contains an essential separating closed surface of genus g. The resulting manifolds M(r) are still hyperbolic. This contrasts sharply with known finiteness results on Dehn filling and with the known finiteness result on handle addition for the cases g = 0,1. Our 3-manifold M is the complement of a hyperbolic, small knot in a handlebody of genus 3.

Hyperbolic knot, small knot, handle additions
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M50
Received: 13 June 2005
Revised: 14 January 2006
Published: 1 January 2007
Ruifeng Qiu
Department of Mathematics
Dalian University of Technology
Dalian 116022
Shicheng Wang
Department of Mathematics
Peking University
Beijing 100871