TY - JOUR

T1 - Normalized entropy versus volume for pseudo-Anosovs

AU - Kojima, Sadayoshi

AU - McShane, Greg

N1 - Funding Information:
Kojima is partially supported by Grant-in-Aid for Scientific Research (A) (No.18204004), JSPS, Japan.

PY - 2018/4/5

Y1 - 2018/4/5

N2 - 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.

AB - 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.

KW - Entropy

KW - Hyperbolic volume

KW - Mapping class

KW - Mapping torus

KW - Teichmüller translation distance

KW - Weil-Petersson metric

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U2 - 10.2140/gt.2018.22.2403

DO - 10.2140/gt.2018.22.2403

M3 - Article

AN - SCOPUS:85045302933

VL - 22

SP - 2403

EP - 2426

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

IS - 4

ER -