2026/07/23 by Francesco Fournier-Facio, Matthew C. B. Zaremsky
#math.GR
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 FP2. One naturally wonders whether analogous results hold for the higher finiteness properties Fn and FPn (3 ≤ n ≤ ∞), i.e., whether every finitely generated recursively presented group embeds into a group of type Fn and whether every finitely generated group embeds into a group of type FPn. We prove that a positive answer to the FPn question that moreover preserves recursive presentability would imply a positive answer to the Fn 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 FP3(ℚ) (hence also never FP3 nor F3). Thus, any approach to the questions of higher finiteness properties must go via a different route than the Higman rope trick.