Recursive simple envelope by cyclic HNN extensions

A finitely generated group with soluble word problem for a group embeds in a countably generated simple group with a recursive presentation of a group. Start with $P_0=G\ast ⟨a⟩$, retaining decidable membership in $⟨a⟩$. At stage $n$, add a free infinite cyclic generator $z_n$. For every nonidentity base element $g$, conjugate $[g,z_n]$ to $a$ and conjugate $g{z}_n$ to $a$; also conjugate $z_n$ to $a$, using distinct stable letters. All associated subgroups are infinite cyclic, with decidable membership by effective cyclic membership in a free product. Britton's lemma preserves embeddings, decidable word problem and decidable membership in the inherited $⟨a⟩$.

In the union, any nontrivial normal subgroup contains some nonidentity $g$ and hence $a$, by the commutator-conjugation relation. The other relations force $z_n$ and then every old $g$ into the normal closure of $a$. Every stable letter becomes an old element at a later stage, so the normal closure of $a$ is the whole union. The union is nontrivial and simple. Effective word enumeration at each stage makes all relators recursively enumerable. A two-generator HNN embedding followed by the Higman embedding theorem embeds this simple envelope in a finitely presented group.