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

Anthony Sanchez (University of California San Diego)

Monday 26th September, 2022 16:00-17:00 Maths 311B

Abstract

Platonic solids have been studied for thousands of years. By unfolding a platonic solid we can  associate to it a translation surface. Interesting information about the underlying platonic solid can be discovered in the cover where more (dynamical and geometric) structure is present. The translation covers we consider have a large group of symmetries that leave the global composition of the surface unchanged. However, the local structure of paths on the surface is often sensitive to these symmetries. The Kontsevich-Zorich mondromy group keeps track of this sensitivity.

In joint work with R. Gutiérrez-Romo and D. Lee, we study the monodromy groups of translation covers of some platonic solids and show that the Zariski closure is a power of SL(2,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").

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