Orateur
Description
A dilation surface is roughly a surface made up of polygons in the complex plane with parallel sides glued together by complex linear maps. The action of SL(2, R) on the plane induces an action of SL(2, R) on the collection of all dilation surfaces, aka the moduli space, which possesses a natural manifold structure. As with translation surfaces, which can be identified with holomorphic 1-forms on Riemann surfaces, dilation surfaces can be thought of as “twisted” holomorphic 1-forms.
I will describe joint work with Nick Salter producing an SL(2, R)-invariant measure on the moduli space of dilation surfaces that is mutually absolutely continuous with respect to Lebesgue measure. The construction fundamentally uses the group cohomology of the mapping class group with coefficients in the homology of the surface. It also relies on joint work with Matt Bainbridge and Jane Wang, showing that the moduli space of dilation surfaces is a K(pi,1) where pi is the framed mapping class group. I will describe further work with Bainbridge and Wang which shows that no such SL(2, R)-invariant measure can be finite. This will follow from showing that an open and dense set of surfaces in a stratum diverge under the action of the diagonal subgroup of SL(2, R) and hence the conclusion of Poincare recurrence fails.
