Finiteness properties and Higman's rope trick

Abstract

In 1961 Higman proved that every finitely generated recursively presented group embeds into a finitely presented group. In 2018 Leary proved that every finitely generated group embeds into a group of type $FP_2$. One naturally wonders whether analogous results hold for the higher finiteness properties $F_n$ and $FP_n$ ($3 \leq n \leq \infty$), i.e., whether every finitely generated recursively presented group embeds into a group of type $F_n$ and whether every finitely generated group embeds into a group of type $FP_n$. We prove that a positive answer to the $FP_n$ question that moreover preserves recursive presentability would imply a positive answer to the $F_n$ question. We also investigate the output groups from Higman's and Leary's proofs, which in both cases arise from the so-called ``Higman rope trick'', and find that they are never of type $FP_3(\mathbb{Q})$ (hence also never $FP_3$ nor $F_3$). Thus, any approach to the questions of higher finiteness properties must go via a different route than the Higman rope trick.