Kontsevich-Zorich monodromy groups of translation covers of some platonic solids

TL;DR

This paper computes the Zariski closure of Kontsevich-Zorich monodromy groups for specific translation covers of platonic solids, showing they are equal to a power of SL(2,R), and also determines their Lyapunov spectra.

Contribution

It provides explicit calculations of monodromy groups for certain translation surfaces and establishes their Zariski closure as a power of SL(2,R), combining algebraic and dynamical methods.

Findings

Zariski closure of monodromy groups equals a power of SL(2,R)

Generators for the monodromy groups are explicitly found

Lyapunov spectra are computed for the surfaces

Abstract

We compute the Zariski closure of the Kontsevich-Zorich monodromy groups arising from certain square tiled surfaces that are geometrically motivated. Specifically we consider three surfaces that emerge as translation covers of platonic solids and quotients of infinite polyhedra, and show that the Zariski closure of the monodromy group arising from each surface is equal to a power of $\rm{SL}(2, \mathbb{R})$. We prove our results by finding generators for the monodromy groups, using a theorem of Matheus-Yoccoz-Zmiaikou that provides constraints on the Zariski closure of the groups (to obtain an "upper bound"), and analyzing the dimension of the Lie algebra of the Zariski closure of the group (to obtain a "lower bound"). Moreover, combining our analysis with the Eskin-Kontsevich-Zorich formula, we also compute the Lyapunov spectrum of the Kontsevich-Zorich cocycle for said square-tiled surfaces.