#### Volume 24, issue 1 (2020)

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$\mathrm{GL}_2 \mathbb{R}$–invariant measures in marked strata: generic marked points, Earle–Kra for strata and illumination

### Paul Apisa

Geometry & Topology 24 (2020) 373–408
##### Abstract

We show that nontrivial $GL\left(2,ℝ\right)$–invariant point markings for translation surfaces in strata of abelian differentials exist only when the translation surface belongs to a hyperelliptic component. As an application, we establish constraints on sections of the universal curve restricted to orbifold covers of subvarieties of the moduli space of Riemann surfaces that contain a Teichmüller disk. We also solve the finite blocking problem for generic translation surfaces.

##### Keywords
dynamics on Teichmüller space, flat surfaces, translation surfaces
##### Mathematical Subject Classification 2010
Primary: 30F60, 32G15, 37F30