Identifying opposite sides of a square yields a torus. Doing this for a regular octagon or other appropriate polygonal shapes in the plane, yields higher genus ``flat" surfaces. The most interesting class in Thurston's classification of homeomorphisms of surfaces are called pseudo-Anosov, or simply pA. Each pA has a flat surface associated to it, as well as a distinguished direction for flow on the surface. Sah and Arnoux-Fathi defined an invariant for certain interval maps that Arnoux showed to give an invariant for flows on flat surfaces. This thus defines an invariant of pA maps.
In work with each of Arnoux, Calta and Do, we showed that this invariant of a pA vanishes exactly when a related algebraic unit (the dilatation of the pA) has a non-palindromic minimal polynomial, and gave various examples. In this talk, we explain some of the above and point to open questions.