有限性性质与希格曼绳技巧
Finiteness properties and Higman's rope trick
浏览论文内容
中文总结 AI 辅助
探讨有限生成递归表示群能否嵌入$F_n$型群以及有限生成群能否嵌入$FP_n$型群，证明对$FP_n$问题肯定回答且保持递归可表示性意味着对$F_n$问题肯定回答，还发现希格曼和利里证明中的输出群非$FP_3(\mathbb{Q})$型。
中文摘要 AI 辅助
1961年希格曼证明每个有限生成递归表示群可嵌入到一个有限表示群中。2018年利里证明每个有限生成群可嵌入到一个$FP_2$型群中。人们自然想知道对于更高的有限性性质$F_n$和$FP_n$（$3\leq n\leq\infty$）类似结果是否成立。我们证明对$FP_n$问题的肯定回答且保持递归可表示性将意味着对$F_n$问题的肯定回答。我们还研究了希格曼和利里证明中的输出群，发现它们都不是$FP_3(\mathbb{Q})$型（因此也不是$FP_3$和$F_3$型）。所以，任何解决更高有限性性质问题的方法都必须不同于希格曼绳技巧。
英文摘要
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.
