We describe a computational, heuristic approach to the problem of deciding whether or not a given finitely presented group has a free quotient of rank two or more. Our strategy is to construct a finite nilpotent quotient of the given group, to search for quotients that are free within a variety containing that quotient, and then lift to the original group. We give theoretical justification to our strategy, and describe successful computations with sections of the Picard group $\SL _2(\Z[i])$.
"Free quotients of finitely presented groups." Experiment. Math. 5 (1) 49 - 56, 1996.