AI 中文总结
该研究刻画了有限生成群中无上下文性子群的判定条件，证明群可嵌入R.汤普森群$V$的等价条件，并指出存在最难的无上下文性成员问题。
AI 中文摘要
考虑$V_*$，即R.汤普森群$V$在对康托尔集$\{0,1\}^\omega$的自然作用下稳定$0^\omega$的子群。设$G_*$为有限生成群$G$的任意子群，我们证明$G_*$是$G$的无上下文性子群当且仅当$G_*$是同态$G \rightarrow V$下$V_*$的拉回。特别地，这表明存在最难的无上下文性成员问题。由此，我们证明群$G$可嵌入$V$当且仅当它是有限个无上下文自动机的并的转移群，等价于存在$G$中有限个无上下文性子群，其核的交集为平凡。
英文摘要
Consider $V_*$, the subgroup of R. Thompson's group $V$ which stabilises $0^ω$ under the natural action upon the Cantor set, $\{0, 1\}^ω$. Let $G_*$ be any subgroup of a finitely generated group $G$. We show that $G_*$ is a context-free subgroup of $G$ if and only if $G_*$ is a pullback of $V_*$ under a homomorphism $G \rightarrow V$. In particular, this shows the existence of a hardest context-free membership problem. As a consequence, we prove that a group $G$ embeds into $V$ if and only if it is the transition group of a finite union of context-free automata, or equivalently, if there exist finitely many context-free subgroups of $G$ whose cores intersect trivially.
Comments21 pages. This second version amends the bibliography, and removes Theorem 5, which was found to be an immediate corollary of Theorem 1.3 in "Obstructions for subgroups of Thompson's group $V$" by Burillo, Cleary and Röver
