# Chi Cheuk Tsang (Université du Québec à Montréal)

Title: Dilatations of pseudo-Anosov maps.

Abstract: A pseudo-Anosov map is a surface homeomorphism that acts with similar dynamics as a hyperbolic element of SL2R on R2. A classical result of Nielsen and Thurston shows that these are surprisingly prevalent among mapping classes of surfaces. The dilatation of a pseudo-Anosov map is a measure of the complexity of its dynamics. It is another classical result that the set of dilatations among all pseudo-Anosov maps defined on a fixed surface has a minimum element. This minimum dilatation can be thought of as the smallest amount of mixing one can perform on the surface while still doing something topologically interesting. The minimum dilatation problem asks for this minimum value. In this talk, we will start by providing some background for pseudo-Anosov maps, in particular explaining how the theory can be viewed from the perspective of outer automorphisms of surface groups. We will then present some recent work on the minimum dilatation problem with Eriko Hironaka, which shows a sharp lower bound for dilatations of fully-punctured pseudo-Anosov maps with at least two puncture orbits.