Mixed quasi-étale surfaces, new surfaces of general type with pg=0 and their fundamental group

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We call a projective surface X mixed quasi-étale quotient if there exists a curve C of genus g(C)≥ 2 and a finite group G that acts on C× C exchanging the factors such that X=(C× C)/G and the map C× C → X has finite branch locus. The minimal resolution of its singularities is called mixed quasi-étale surface. We study the mixed quasi-étale surfaces under the assumption that (C× C)/G⁰ has only nodes as singularities, where G⁰ G is the index two subgroup of the elements that do not exchange the factors. We classify the minimal regular surfaces with pg=0 whose canonical model is a mixed quasi-étale quotient as above. All these surfaces are of general type and as an important byproduct, we provide an example of a numerical Campedelli surface with topological fundamental group ₄, and we realize 2 new topological types of surfaces of general type. Three of the families we construct are -homology projective planes.