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

Olga Kharlampovich; Rizos Sklinos

doi:10.4171/ggd/872

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

Olga Kharlampovich
Hunter College and Graduate Center, CUNY, New York, USA
Rizos Sklinos
Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing, P. R. China

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

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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 $σ$ (in the language of groups) if and only if $σ$ 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 $d < 1/2$ .

Cite this article

Olga Kharlampovich, Rizos Sklinos, First-order sentences in random groups II: $\forall\exists$ -sentences. Groups Geom. Dyn. (2025), published online first

DOI 10.4171/GGD/872