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

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 $g$.