Rank and genus of 3-manifolds

Li, Tao

doi:10.1090/S0894-0347-2013-00767-5

Rank and genus of 3-manifolds
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by Tao Li

J. Amer. Math. Soc. 26 (2013), 777-829

DOI: https://doi.org/10.1090/S0894-0347-2013-00767-5

Published electronically: February 27, 2013

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Abstract:

We construct a counterexample to the Rank versus Genus Conjecture, i.e. a closed orientable hyperbolic 3-manifold with rank of its fundamental group smaller than its Heegaard genus. Moreover, we show that the discrepancy between rank and Heegaard genus can be arbitrarily large for hyperbolic 3-manifolds. We also construct toroidal such examples containing hyperbolic JSJ pieces.

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