We study subsets $E$ of finitely generated groups where the set of all words over a given finite generating set that lie in $E$ forms a context-free language. We call these sets recognisably context-free. They are invariant of the choice of generating set and a theorem of Muller and Schupp fully classifies when the set $\{1\}$ can be recognisably context-free. We extend Muller and Schupp's result to show that a group $G$ admits a finite recognisably context-free subset if and only if $G$ is virtually free. We show that every conjugacy class of a group $G$ is recognisably context-free if and only if $G$ is virtually free. We conclude by showing that a coset is recognisably context-free if and only if the Schreier coset graph of the corresponding subgroup is quasi-isometric to a tree.

中文翻译：

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具有上下文无关原像的组的子集

我们研究有限生成组的子集 $E$，其中位于 $E$ 中的给定有限生成集上的所有单词的集合形成上下文无关语言。我们称这些集合为可识别的上下文无关的。它们对于生成集的选择是不变的，并且当集合 $\{1\}$ 可以识别为上下文无关时，Muller 和 Schupp 定理可以完全分类。我们扩展了 Muller 和 Schupp 的结果，表明群 $G$ 承认一个有限的可识别的上下文无关子集当且仅当 $G$ 几乎是自由的。我们证明群 $G$ 的每个共轭类都是可识别的上下文无关当且仅当 $G$ 几乎是自由的。我们的结论是，当且仅当相应子群的 Schreier 陪集图与树拟等距时，陪集是可识别的上下文无关的。