Arithmeticity of the Kontsevich--Zorich monodromies of certain families of square-tiled surfaces

TL;DR

This paper investigates the frequency of arithmetic groups within the Kontsevich--Zorich monodromies associated with square-tiled surfaces of low genus, revealing that such groups are common in these cases.

Contribution

It demonstrates that arithmetic groups frequently occur among the Kontsevich--Zorich monodromies of low-genus square-tiled surfaces, expanding understanding of their algebraic properties.

Findings

Arithmetic groups are common among low-genus Kontsevich--Zorich monodromies.

The study provides new insights into the structure of monodromies associated with square-tiled surfaces.

Results suggest a prevalence of arithmeticity in these monodromies for certain families.

Abstract

The variations of Hodge structures of weight one associated to square-tiled surfaces naturally generate interesting subgroups of integral symplectic matrices called Kontsevich--Zorich monodromies. In this paper, we show that arithmetic groups are frequent among the Kontsevich--Zorich monodromies of square-tiled surfaces of low genera $g$.