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arXiv · 2406.12697

The small boundary property in products

David Kerr·Hanfeng Li
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Abstract

For a continuous action $G\curvearrowright X$ of a countable group on a compact metrizable space we show that the following are equivalent: (i) the action $G\curvearrowright X$ has the small boundary property and no finite orbits, (ii) for every continuous action $H\curvearrowright Y$ of a countable group on a compact metrizable space, the product action $G\times H\curvearrowright X\times Y$ has the small boundary property. In particular, (ii) is automatic when $G$ is infinite and the action $G\curvearrowright X$ is minimal and has the small boundary property. The argument relies on a small boundary version of the Urysohn lemma.

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David Kerr, Hanfeng Li. 2024-06-18. The small boundary property in products. https://arxiv.org/abs/2406.12697

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