Formal solutions and the first-order theory of acylindrically hyperbolic groups

We generalise Merzlyakov’s theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if $G$ is an acylindrically hyperbolic group and $E(G)$ denotes the unique maximal finite normal subgroup of $G$, then $G$ and the HNN extension $G{\dot{\ast }}_{E(G)}$, which is simply the free product $G\ast \mathbb{Z}$ when $E(G)$ is trivial, have the same $\mathrm{\forall }\mathrm{\exists }$-theory. As a consequence, we prove the following conjecture, formulated by Casals-Ruiz, Garreta and de la Nuez Gonz'alez: acylindrically hyperbolic groups have trivial positive theory. In particular, one recovers a result proved by Bestvina, Bromberg and Fujiwara, stating that, with only the obvious exceptions, verbal subgroups of acylindrically hyperbolic groups have infinite width.