AI 中文总结
该研究证明了n≥2时Brin-Thompson群$nV$中元素的拓扑共轭性不可判定，还得到了Cantor空间同胚群有限生成超群中Whitehead问题及群论共轭的不可判定性。
AI 中文摘要
我们证明，对于每个n≥2，Brin-Thompson群$nV$中元素的拓扑共轭性是不可判定的。从图灵机$T$出发，归约过程会生成$nV$中的两个元素：若$T$在空带上停机，则这两个元素在$nV$中被一个对合共轭；若$T$不在空带上停机，则这两个群元素的拓扑动力学可通过它们对（芽）非周期点集合的极小子系统的访问情况来区分。在这些群以及Cantor空间同胚群内的任何有限生成超群中，我们还得到了Whitehead问题（即处于同一自同构轨道的问题）的不可判定性，以及群论共轭的不可判定性。
英文摘要
We prove that topological conjugacy of elements of the Brin--Thompson group $nV$ is undecidable for every $n\geq2$. From a Turing machine $T$, the reduction produces two elements of $nV$, which are conjugate by an involution in $nV$ if $T$ halts on the empty tape. If $T$ does not halt on the empty tape, the topological dynamics of the two group elements are distinguished by their visits to minimal subsystems of the set of (germ-)aperiodic points. In these groups, and any finitely-generated supergroups inside the homeomorphism group of Cantor space, we also obtain undecidability of the Whitehead problem (the problem of being in the same automorphism orbit), and undecidability of group-theoretic conjugacy.
Comments17 pages
