The realization problem for twisted quadratic differentials (dilation surfaces)

Twisted quadratic differentials, also known as dilation surfaces, are geometric structures that are in a way a generalization of translation surfaces. We can define a dilation surface either as a quadratic differential twisted by some real holonomy or as a collection of polygons with sides identified by translations and dilations by nonzero real factors. This small generalization is enough to introduce interesting new dynamical behaviors on dilation surfaces that do not occur for translation surfaces. In this talk, we will introduce dilation surfaces and discuss some of the new and interesting dynamical behaviors that can occur on them. We will then formulate the realization problem, which asks which mapping class group elements can be realized as affine automorphisms of a dilation surfaces, and discuss challenges and progress toward resolving this problem.

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