# Pseudo-Anosov maps with small dilatation

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Joan Birman, Columbia University
Fine Hall 314

Fix an orientable surface $S$. It is known that the set of dilatations of all pseudo-Anosov maps acting on $S$ is a family of real numbers that is bounded below by 1, and has a minimum value $\lambda_{min,S}>1$ which is realized geometrically. We will discuss recent work on the problem of determining $\lambda_{min,S}$ and show how a little-known theorem, the 'Coefficient Theorem for Digraphs,' can be used to gain insight into this set. The study of small dilatation pA maps appears to be related to the study of small volume fibered hyperbolic 3-manifolds, and an example from 3-manifolds has played a role in understanding the dilatation problem.