Topological and dynamical properties of pseudo-Anosov maps vs. their action on homology

Abstract

Let $S$ be a surface of finite type and $\Mod(S)$ be its mapping class group. For a mapping class $f \in \Mod(S)$, we can consider its action on $H_1(S

\Z)$. Broadly speaking, we study the interaction between the dimension of the fixed subspace of this action, denoted by $\kappa(f)$, and other properties of $f$. More precisely, we first consider the notion of Torelli groups of subsurfaces introduced by Putman and we compute bounds for their cohomological dimensions and their minimal asymptotic translation lengths on the curve graph of the surface. We then turn our attention to the study of the relation between the entropy of a pseudo-Anosov map $f$ and the quantity $\kappa(f)$. This has been studied by Agol-Leininger-Margalit for closed surfaces and we partially generalize their work to punctured surfaces.