Semisimplicity and Rigidity of the Kontsevich–Zorich cocycle for products of strata

Abstract. The moduli space of translation surfaces carries an action of ( \mathrm{SL}2\mathbb{R} ) that is central to understanding dynamics on individual surfaces. We generalize this to actions on *products of translation surfaces* by a subgroup ( G \le \prod{i=1}^n \mathrm{SL}_2\mathbb{R} ). Using variations of Hodge structure, we show that invariant subbundles for the (G)-action respect the Hodge structure, yielding semisimplicity results. Our key result is a joint polynomiality theorem: for any affine invariant manifold ( \mathcal{M} ) inside a product of strata, the (G)-equivariant orthogonal projector onto a flat invariant subbundle (and its tensors) has, in period/affine coordinates ( (x,y) ), entries rational in ( (x,y) ) with a fixed quadratic denominator ( A(x,y) ); equivalently, the entries are homogeneous of degree (0) in (x,y,1/A).

Recommended citation: Polina Baron. (2025). "Semisimplicity and Rigidity of the Kontsevich–Zorich cocycle for products of strata." In preparation.
Download Slides