Generalized Burniat Type surfaces and Bagnera-de Franchis Varieties

In this article we construct three new families of surfaces of general type with pg = q = 0,K2 = 6, and seven new families of surfaces of general type with pg = q = 1,K2 = 6, realizing 10 new fundamental groups. We also show that these families correspond to pairwise distinct irreducible connected components of the Gieseker moduli space of surfaces of general type. We achieve this using two different main ingredients. First we introduce a new class of surfaces, called generalized Burniat type surfaces, and we completely classify them (and the connected components of the moduli space containing them). Second, we introduce the notion of Bagnera-de Franchis varieties: these are the free quotients of an Abelian variety by a cyclic group (not consisting only of translations). For these we develop some basic results