共轭子长度与等周函数
Conjugator Lengths and Isoperimetric functions
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中文总结 AI 辅助
该研究构造了有限表示群，揭示共轭子长度函数与Dehn函数这两个不变量具有强独立性，还给出了共轭子长度函数递归但字、共轭问题均不可判定的首个例子。
中文摘要 AI 辅助
我们证明，对于任意具有超加性时间函数f的识别S-机器（recognizing S-machine）S，存在一个有限表示群，其共轭子长度函数为二次，且其Dehn函数增长速度与f的平方相当。该结果与作者此前一篇论文的结果对偶，此前论文构造了具有三次Dehn函数且拥有各类共轭子长度函数的群族。由此，我们给出了首个已知的有限表示群例子，其共轭子长度函数是递归的，但字问题与共轭问题均不可判定，这回答了Rips问题的一个类似问题。此外，给定一个字问题可判定的有限表示群G，我们得到一个共轭问题可判定的有限表示群，其Dehn函数增长速度快于G的Dehn函数。最后，将该结果与其对偶结果结合，对于大量函数对(f,g)，我们提供了一个有限表示群的例子，其Dehn函数等价于f，且共轭子长度函数等价于g。这表明这两个不变量具有极强的独立性，为Bridson、Riley与Sale的问题取得了重大进展。
英文摘要
We show that for any recognizing $S$-machine $\textbf{S}$ with superadditive time function $f$, there exists a finitely presented group whose conjugator length function is quadratic and whose Dehn function grows like the square of $f$. This result is dual to that of a previous paper of the authors, which constructed a family of groups with cubic Dehn function that have various conjugator length functions. We thereby give the first known example of a finitely presented group whose conjugator length function is recursive, but whose Word and Conjugacy Problems are both undecidable. This answers an analogue of a question of Rips. Moreover, given a finitely presented group $G$ with decidable Word Problem, we obtain a finitely presented group with decidable Conjugacy Problem whose Dehn function grows faster than that of $G$. Finally, combining this result with its dual, for a wide array of pairs of functions $(f,g)$ we furnish an example of a finitely presented group with Dehn function equivalent to $f$ and conjugator length function equivalent to $g$. This shows that the two invariants are very strongly independent and making significant progress on a question of Bridson, Riley, and Sale.
