Normalized entropy versus volume for pseudo-Anosovs

Sadayoshi Kojima, Greg McShane

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


Thanks to a recent result by Jean-Marc Schlenker, we establish an explicit linear inequality between the normalized entropies of pseudo-Anosov automorphisms and the hyperbolic volumes of their mapping tori. As corollaries, we give an improved lower bound for values of entropies of pseudo-Anosovs on a surface with fixed topology, and a proof of a slightly weaker version of the result by Farb, Leininger and Margalit first, and by Agol later, on finiteness of cusped manifolds generating surface automorphisms with small normalized entropies. Also, we present an analogous linear inequality between the Weil-Petersson translation distance of a pseudo-Anosov map (normalized by multiplying by the square root of the area of a surface) and the volume of its mapping torus, which leads to a better bound.

Original languageEnglish
Pages (from-to)2403-2426
Number of pages24
JournalGeometry and Topology
Issue number4
Publication statusPublished - 2018 Apr 5
Externally publishedYes


  • Entropy
  • Hyperbolic volume
  • Mapping class
  • Mapping torus
  • Teichmüller translation distance
  • Weil-Petersson metric

ASJC Scopus subject areas

  • Geometry and Topology


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