Entropy versus volume via Heegaard diagrams

Abstract

The following inequalities are established, improving a former inequality due to Kojima. For any closed arithmetic hyperbolic $3$--manifold fibering over a circle, the entropy of the pseudo-Anosov monodromy is bounded by the hyperbolic volume of the $3$--manifold, up to a universal constant factor. For any closed hyperbolic $3$--manifold fibering over a circle with systole $\ge \epsilon >0$, the entropy is bounded by the hyperbolic volume times $\log (3+1/\epsilon )$, up to a universal constant factor. The proof relies on Heegaard Floer homology and hyperbolic geometry.

52 pages, 1 figure; Corollary 1.3 and Figure 1 added; minor revisions