Topological entropy of pseudo-Anosov maps on punctured surfaces vs. homology 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 $\frac{(k+1)\log (k+3)}{|\chi (S)|}$ 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.

20 pages, 8 figures, comments are welcome