On the fundamental group of self-affine plane tiles

Let A epsilon Z(2) x Z(2) be an expanding matrix, D subset of Z(2) a set with vertical bar det(A)vertical bar elements and define T via the set equation AT = T + D. If the two-dimensional Lebesgue measure of T is positive we call T a self-affine plane tile. In the present paper we are concerned with topological properties of T. We show that the fundamental group pi(1) (T) of T is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of pi(1) (T). Furthermore, we give a short proof of the fact that the closure of each component of int(T) is a locally connected continuum (we prove this result even in the more general case of plane IFS attractors fulfilling the open set condition). If pi(1)(T) = 0 we even show that the closure of each component of int(T) is homeomorphic to a closed disk. We apply our results to several examples of tiles which are studied in the literature.