Rank and genus of 3-manifolds

2011/06/30 by Tao Li, Li, Tao · 1 citation
Mathematics · #57M05 #57M27 #57M50 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #math.GR #math.GT #msc:57M05 #msc:57M27 #msc:57M50

paper · pdf · doi:10.48550/arxiv.1106.6302

60 pages, 15 figures. Minor changes, typos corrected

openalex publication_date 2011/06/30 · arxiv created 2013/01/23 · arxiv updated 2013/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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