Geometry and rank of fibered hyperbolic 3–manifolds

Biringer, Ian

doi:10.2140/agt.2009.9.277

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Geometry and rank of fibered hyperbolic

$3$ –manifolds

Ian Biringer

Algebraic & Geometric Topology 9 (2009) 277–292

DOI: 10.2140/agt.2009.9.277

Abstract

Recall that the rank of a finitely generated group is the minimal number of elements needed to generate it. In [Comm. Anal. Geom. 10 (2002) 377-395], M White proved that the injectivity radius of a closed hyperbolic $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn></math>$ –manifold $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>M</mi></math>$ is bounded above by some function of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> rank</mo><mrow><mo class="MathClass-open">(</mo><mrow><msub><mrow><mi>π</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>M</mi></mrow><mo class="MathClass-close">)</mo></mrow></mrow><mo class="MathClass-close">)</mo></mrow></math>$ . Building on a technique that he introduced, we determine the ranks of the fundamental groups of a large class of hyperbolic $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn></math>$ –manifolds fibering over the circle.

Keywords

rank, fundamental group, hyperbolic

$3$ -manifold

Mathematical Subject Classification 2000

Primary: 57M50

References

Publication

Received: 11 June 2008

Revised: 7 September 2008

Accepted: 19 January 2009

Published: 13 February 2009

Authors

Ian Biringer
Department of Mathematics University of Chicago Chicago, IL 60637 USA
http://www.math.uchicago.edu/~biringer

Volume 9, issue 1 (2009)