Topologically transitive homeomorphisms of quotients of tori

This paper considers the following question: for what finite subgroups G ⊂ GL(n, ℤ), does there exist an element A ∈ GL(n, ℤ) inducing a topologically transitive homeomorphism of double-struck T signn/G? We show that for n = 2 and 3, the only possibility is G = {±I}. Curiously, in higher dimension the structure is less restrictive. We give a variety of examples in dimension 4. Nevertheless, we show that in dimension ≥ 4, there are relatively few irreducible examples.