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

TL;DR

This paper investigates the dynamics of linear flows on infinite periodic covers of Veech surfaces, revealing that certain Veech groups have large kernels in their homological representations, with applications to wind-tree models.

Contribution

It demonstrates that the kernel of natural homological representations of Veech groups on infinite covers is large, answering a question by Pascal Hubert.

Findings

The kernel of some Veech group representations is 'big' in size.

Results apply to dynamics on wind-tree models with Veech surfaces.

Provides new insights into the structure of Veech groups for infinite covers.

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.