On threefolds isogenous to a product of curves

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(OX) 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(OX)=−1, under the assumption that the group acts faithfully on each factor.