Computable Scott sentences and the weak Whitehead problem for finitely presented groups

Abstract

We prove that if A is a computable Hopfian finitely presented structure, then A has a computable d-Σ2 Scott sentence if and only if the weak Whitehead problem for A is decidable. We use this to infer that every hyperbolic group as well as any polycyclic-by-finite group has a computable d-Σ2 Scott sentence, thus covering two main classes of finitely presented groups. Our proof also implies that every weakly Hopfian finitely presented group is strongly defined by its ∃+-types, a question which arose in a different context.