An Incoherent Simple Group

Proposition 1. (Grunewald [5, Proposition B]) Let F be a free group of rank k ≥ 2, generated by {s1, . . . , sk}. Let r1, . . . , rm be words over {s1, . . . , sk} ±1 and R their normal closure 〈〈r1, . . . , rm〉〉F in F . Let H be the group with presentation 〈s1, . . . , sk | r1, . . . , rm〉 and φ the canonical epimorphism φ : F → H ∼= F/R. Let F be a free group of rank k generated by {t1, . . . , tk} and ψ the isomorphism F → F , mapping si to ti, i = 1, . . . , k. Let H = 〈t1, . . . , tk | ψ(r1), . . . , ψ(rm)〉, and ψ̃ : H → H the isomorphism induced by ψ. Finally, let φ be the canonical epimorphism F → H ∼= F/ψ(R) (see the commutative diagram below for a summary of this notation). Suppose that H is infinite and R 6= {1}. Then the group {(s, t) ∈ F × F : ψ̃(φ(s)) = φ(t)} is a subgroup of F × F generated by the k +m elements (s1, t1), . . . , (sk, tk), (r1, 1), . . . , (rm, 1), but it is not finitely presented.