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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\text{SL}(2,{\mathbb{R}}). We prove our results by finding generators for the monodromy groups, using a theorem of Matheus–Yoccoz–Zmiaikou (2014) 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 (2014), we also compute the Lyapunov spectrum of the Kontsevich–Zorich cocycle for said square-tiled surfaces.
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An arithmetic Kontsevich–Zorich monodromy of a symmetric origami in genus 4
We demonstrate the existence of a certain genus four origami whose Kontsevich--Zorich monodromy is arithmetic in the sense of Sarnak. The surface is interesting because its Veech group is as large as possible and given by SL(2,Z). When compared to other surfaces with Veech group SL(2,Z) such as the Eierlegendre Wollmichsau and the Ornithorynque, an arithmetic Kontsevich--Zorich monodromy is surprising and indicates that there is little relationship between the Veech group and monodromy group of origamis. Additionally, we record the index and congruence level in the ambient symplectic group which gives data on what can appear in genus 4.
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- Award ID(s):
- 2103136
- PAR ID:
- 10639734
- Publisher / Repository:
- New York Journal of Mathematics
- Date Published:
- Journal Name:
- New York journal of mathematics
- ISSN:
- 1076-9803
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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