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 $\mathrm{SL}(2, \mathbb R)$ on the plane induces an action of $\mathrm{SL}(2, \mathbb R)$ 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.

The first result that I will present, joint with Nick Salter, produces an $\mathrm{SL}(2, \mathbb R)$-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. 

The second result that I will present, joint with Bainbridge and Wang, is that an open and dense set of dilation surfaces have Morse-Smale dynamics, i.e. the horizontal straight lines spiral towards a finite set of simple closed curves in their forward and backward direction. A consequence is that any fully supported $\mathrm{SL}(2, \mathbb R)$ invariant measure on the moduli space of dilation surfaces cannot be a finite measure.