Every countable group embeds in a group of type ${\mathrm{FP}}_n$

Abstract

For every integer $n\ge 2$, we prove that every countable group embeds in a group of type ${\mathrm{FP}}_n$. We also construct a group of type ${\mathrm{F}}_n$ containing a copy of every recursively presented group. Consequently, a finitely generated group embeds in a group of type ${\mathrm{F}}_n$ if and only if it is recursively presented. This answers questions of Fournier-Facio and Zaremsky, and confirms a suggestion of Gromov. An appendix combines the construction with the controlled-filling argument in a subsequently released OpenAI preprint to prove that every countable group embeds in a two-generator group of type ${\mathrm{FP}}_{\infty }$.

Produced with the help of ChatGPT Astra