Topological entropy of pseudo-Anosov maps on punctured surfaces vs. homology of mapping tori

TL;DR

This paper explores the relationship between the topological entropy of pseudo-Anosov maps on punctured surfaces and the first homology of their mapping tori, providing bounds and generalizations of previous results.

Contribution

It establishes an upper bound on entropy in terms of homology rank for pseudo-Anosov maps on punctured surfaces, extending prior work by Tsai and Agol-Leininger-Margalit.

Findings

Entropy is bounded by a function of homology rank and surface characteristics.

Provides a partial generalization of previous entropy bounds.

Connects topological entropy with algebraic invariants of mapping tori.

Abstract

We investigate the relation between the topological entropy of pseudo-Anosov maps on surfaces with punctures and the rank of the first homology of their mapping tori. On the surface $S$ of genus $g$ with $n$ punctures, we show that the entropy of a pseudo-Anosov map is bounded from above by $\dfrac{(k+1)\log(k+3)}{|\chi(S)|}$ up to a constant multiple when the rank of the first homology of the mapping torus is $k+1$ and $k, g, n$ satisfy a certain assumption. This is a partial generalization of precedent works of Tsai and Agol-Leininger-Margalit.