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The conjecture is something like the following:

The minimal dilatation among pseudo-Anosov mapping classes on a surface $S_{g,n}$ is realized by $\rho\circ\omega$ where $\omega$ is supported on a subsurface and $\rho$ is finite order.

I was wondering if anyone knew of any work towards this conjecture or a reference for a statement of it.

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[Update]

Margalit has been stating a more precise version of this conjecture since at least 2012. See page 38 of these slides.

[Old version]

There is related work by several people. I could not find a statement of the conjecture, but I have heard versions of it - sometimes it is generalized to more than just the minimal dilatation pA map. Instead, there is a statement about all pA maps satisfying certain bounds. Here are a few relevant references.

  • Small dilatation pseudo-Anosovs and 3–manifolds - by Benson Farb, Christopher J. Leininger, and Dan Margalit.
  • Ideal triangulations of pseudo-Anosov mapping tori - by Ian Agol.
  • Penner sequences and asymptotics of minimum dilatations for subfamilies of the mapping class group - by Eriko Hironaka
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