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)/G 0 has only nodes as singularities, where G 0 ⊳ 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 Z4, and we realize 2 new topological types of surfaces of general type. Three of the families we construct are Q-homology projective planes.

Let be a smooth projective curve and be a finite subgroup of Aut( ) 2 ⋊ ℤ 2 whose action is mixed, i.e. there are elements in exchanging the two isotrivial fibrations of × . Let 0 ⊲ be the index two subgroup ∩ Aut( ) 2 . If 0 acts freely, then ∶= ( × )∕ is smooth and we call it semi-isogenous mixed surface. In this paper we give an algorithm to determine semi-isogenous mixed surfaces with given geometric genus, irregularity and self-intersection of the canonical class. As an application we classify irregular semi-isogenous mixed surfaces with 2 > 0 and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with = 1. We provide an example of a minimal surface of general type with 2 = 7 and = = 2. K E Y W O R D SSurfaces of general type, finite group actions M S C ( 2 0 1 0 ) 14J29, 14L30, 14Q10

Abstract. We give a survey on the fundamental group of surfaces isogenous to a higher product. If the surfaces are regular, e.g. if they are Beauville surfaces, the first homology group is a finite group. We present a MAGMA script which calculates the first homology groups of regular surfaces isogenous to a product.

A mixed quasi-étale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely outside a finite subset. A mixed quasi-étale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-étale surfaces with given geometric genus, irregularity and self-intersection of the canonical class. We prove that all irregular mixed quasi-étale surfaces of general type are minimal. As an application, we classify all irregular mixed quasi-étale surfaces of general type with genus equal to the irregularity, and all the regular ones with K 2 > 0, thus constructing new examples of surfaces of general type with χ = 1. We mention the first example of a minimal surface of general type with p g = q = 1 and Albanese fibre of genus bigger than K 2 .

Abstract.A threefold isogenous to a product of curves X is a quotient of a product of three compact Riemann surfaces of genus at least two by the free action of a finite group. In this paper we study these threefolds under the assumption that the group acts diagonally on the product. We show that the classification of these threefolds is a finite problem, present an algorithm to classify them for a fixed value of χ(O X ) and explain a method to determine their Hodge numbers. Running an implementation of the algorithm we achieve the full classification of threefolds isogenous to a product of curves with χ(O X ) = −1, under the assumption that the group acts faithfully on each factor.