First-order sentences in random groups II: $\forall \exists$-sentences

Abstract

We prove that a random group, in Gromov’s density model with $d<1/16$, satisfies with overwhelming probability a universal-existential first-order sentence $\sigma$ (in the language of groups) if and only if $\sigma$ is true in a nonabelian free group. It is remarked that one can also obtain the optimal result, that is, replace $d<1/16$ by $d<1/2$.