Normalized Entropy versus Volume

TL;DR

This paper establishes a linear inequality linking normalized entropies of pseudo-Anosov automorphisms to the hyperbolic volumes of their mapping tori, leading to improved bounds and finiteness results.

Contribution

It introduces a new linear inequality connecting entropy and volume, improving bounds on entropy values and proving finiteness of certain cusped manifolds.

Findings

Derived an explicit linear inequality between normalized entropy and volume.

Provided improved lower bounds for entropy of pseudo-Anosov automorphisms.

Proved a finiteness result for cusped manifolds with small normalized entropy.

Abstract

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 its 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 the square root of the area of a surface) and the volume of its mapping torus, which leads to a better bound.