The following question is proposed by Martino, Tointon, Valiunas and Ventura in [4, question 1·20]:

Let G be a compact group, and suppose that \[\mathcal{N}_k(G) = \{(x_1,\dots,x_{k+1}) \in G^{k+1} \;|\; [x_1,\dots, x_{k+1}] = 1\}\] has positive Haar measure in $G^{k+1}$ . Does G have an open k-step nilpotent subgroup?

We give a positive answer for $k = 2$ .

中文翻译：

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具有一组满足幂零定律的正 Haar 测度的紧群

以下问题由 Martino、Tointon、Valiunas 和 Ventura 在 [4，问题 1·20] 中提出：

令G为紧群，并假设 \[\mathcal{N}_k(G) = \{(x_1,\dots,x_{k+1}) \in G^{k+1} \;|\ ; [x_1,\dots, x_{k+1}] = 1\}\] 在 $G^{k+1}$ 中具有正 Haar 度量。G是否有一个开k步幂零子群？

我们对$k = 2$ 给出肯定的答案。