Conditional decision problems in group theory

It is well known that the triviality problem for finitely presented groups is unsolvable; we ask the question of whether there exists a general procedure to produce a non-trivial element from a finite presentation of a non-trivial group. If not, then this would resolve an open problem by J. Wiegold: “Is every finitely generated perfect group the normal closure of one element?'' We resolve our main question (and several close variants) in the class of finitely generated recursive presentations: there is no algorithm for this class that, on input of a presentation of a non-trivial group, outputs a non-trivial element. We prove a weakened version of our main question: there is no general procedure to output a non-trivial generator from a finite presentation of a non-trivial group. We also show there is neither a general procedure to decompose a finite presentation of a non-trivial free product into two non-trivial finitely presented factors. We apply our results to show that a construction by Stallings on splitting groups with more than one end can never be made algorithmic, nor can the process of splitting connect sums of non-simply connected closed 4-manifolds. The Adian-Rabin construction is used heavily in our proofs. We show that it can be modified in the case of surjections, and that the following close variant is equivalent to our main question: "Is there an algorithm that, on input of a finite presentation of a group, outputs a word which represents the trivial element if and only if the group is trivial?" We define a group to be 'finitely annihilated' if it is the set-theoretic union of its proper finite index subgroups. We show that for several well-understood classes of finitely generated groups, this is equivalent to having non-cyclic abelianisation. We construct an infinite family of finitely presented groups which do not satisfy this criteria, and draw a connection between this property and the Wiegold problem. We apply these results to show that the weig