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

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 $g$.

75 pages, 21 figures, 28 tables, to appear in Transactions of the American Mathematical Society