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Normalized entropy versus volume for pseudo-Anosovs

Sadayoshi Kojima and Greg McShane

Geometry & Topology 22 (2018) 2403–2426

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.

mapping class, entropy, mapping torus, Weil-Petersson metric , Teichmüller translation distance, hyperbolic volume
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 37E30
Received: 9 December 2016
Revised: 16 May 2017
Accepted: 23 June 2017
Published: 5 April 2018
Proposed: Benson Farb
Seconded: Dmitri Burago, Ian Agol
Sadayoshi Kojima
Department of Mathematical and Computing Sciences
Tokyo Institute of Technology
Greg McShane
UFR de Mathématiques
Institut Fourier
St Martin d’Hères