We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

中文翻译：

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无限呈现的置换稳定群和metabelian群的不变随机子群

我们证明了点灯者群L的所有不变随机子群都是共质的。因此，L是置换稳定的，提供了一个无限呈现的此类群的示例。我们的证明更普遍地适用于有限生成的阿贝尔群的所有置换花环产品。我们依赖于顺从群的逐点遍历定理。