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Mixed quasi-étale quotients with arbitrary singularities

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 out of 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 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.