A remark on $\mathbb{Z}^d$-covers of Veech surfaces

Abstract

In this note we are interested in the dynamics of the linear flow on infinite periodic $\mathbb{Z}^d$-covers of Veech surfaces. An elementary remark allows us to show that the kernel of some natural representations of the Veech group acting on homology is "big". In particular, the same is true for the Veech group of the infinite surface, answering a question of Pascal Hubert. We give some applications to the dynamics on wind-tree models where the underlying compact translation surface is a Veech surface.