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18.71 (2014)

Open

(N. Aronszajn). Suppose that $W(x, y) = 1$ in an open subset of $G \times G$, where $G$ is a connected topological group. Must $W(x, y) = 1$ for all $(x, y) \in G \times G$?

Contributor: J. Mycielski

Progress

For locally compact groups the answer is yes.

*No, not necessarily: for any odd integer $n > 10^{10}$ there is a connected topological group such that the identity $x^n = 1$ holds in some neighborhood of unity, but not in the entire group (E. Reznichenko, I. Zyabrev, Preprint, 2024, https://arxiv.org/abs/2406.05203).

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