Entropy versus volume via Heegaard diagrams

Liu, Yi

doi:10.2140/gt.2026.30.1515

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ISSN (electronic): 1364-0380

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Entropy versus volume via Heegaard diagrams

Yi Liu

Geometry & Topology 30 (2026) 1515–1573

DOI: 10.2140/gt.2026.30.1515

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 $\geq 𝜀 > 0$ , the entropy is bounded by the hyperbolic volume times $\log (3 + 1 ∕ 𝜀)$ , up to a universal constant factor. The proof relies on Heegaard Floer homology and hyperbolic geometry.

Keywords

hyperbolic 3-manifold, surface bundle, Heegaard Floer homology

Mathematical Subject Classification

Primary: 57K20, 57K32

Secondary: 57K18

References

Publication

Received: 8 October 2024

Revised: 29 August 2025

Accepted: 15 December 2025

Published: 9 July 2026

Proposed: David Gabai

Seconded: Leonid Polterovich, Gang Tian

Authors

Yi Liu
Beijing International Center for Mathematical Research Peking University Beijing China

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Volume 30, issue 4 (2026)