Low dilatation pseudo-Anosovs on punctured surfaces and volume

TL;DR

This paper demonstrates that for punctured surfaces, pseudo-Anosov homeomorphisms with minimal entropy can have arbitrarily large volume, contrasting the fixed-genus case where volume remains bounded.

Contribution

It constructs examples of pseudo-Anosov homeomorphisms on punctured surfaces with minimal entropy but unbounded volume, showing a fundamental difference from the closed surface case.

Findings

Minimal entropy for punctured surfaces is   

Constructed pseudo-Anosov homeomorphisms with entropy    and volume tending to infinity

Contrasts the fixed-genus case where volume is bounded regardless of entropy

Abstract

For a pseudo-Anosov homeomorphism $f$ on a closed surface of genus $g\geq 2$, for which the entropy is on the order $\frac{1}{g}$ (the lowest possible order), Farb-Leininger-Margalit showed that the volume of the mapping torus is bounded, independent of $g$. We show that the analogous result fails for a surface of fixed genus $g$ with $n$ punctures, by constructing pseudo-Anosov homeomorphism with entropy of the minimal order $\frac{\log n}{n}$, and volume tending to infinity.