Galois structures on plane compacts

In this paper we use results from V. Pasol et al. [Math. Z. 238 (1) (2001) 101–114] and A. Popescu et al. [Math. Nachr., submitted for publication] to construct unusual actions of the absolute Galois group of on a large class of compact subsets of the complex plane. We describe the compact subsets of which admit such a Galois action and develop new techniques to study the connection between the arithmetic of the absolute Galois group of and the topology of .