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arXiv · 2610.18070

Groups with the fixed-point property

Nicolas Monod·Omer Tamuz·Lizzy Teryoshin
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Abstract

We propose the study of the class of groups that fix a point for every continuous action on any convex compact space. Oliver completed the corresponding program of classifying the finite groups that have this fixed-point property in finite dimensions. Turning to infinite groups, the Schauder--Tychonoff theorem establishes that the integers have the fixed-point property, but this has not been shown for any other infinite, finitely generated group. We establish a local transfer principle and use it to show, as a proof of concept, that $\mathbf{Z} \times (\mathbf{Z}/p\mathbf{Z})$ belongs to this class of groups.

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Nicolas Monod, Omer Tamuz, Lizzy Teryoshin. 2026-10-13. Groups with the fixed-point property. https://arxiv.org/abs/2610.18070

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