PreprintCC BY 4.0arXiv
Groups with the fixed-point property
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 Z × (Z/pZ) belongs to this class of groups.
Related
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BibTeX
@misc{monod2026groups,
title = {{Groups with the fixed-point property}},
author = {Nicolas Monod and Omer Tamuz and Lizzy Teryoshin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18070},
eprint = {2610.18070},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18070},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Groups with the fixed-point property
AU - Nicolas Monod
AU - Omer Tamuz
AU - Lizzy Teryoshin
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18070
UR - https://arxiv.org/abs/2610.18070
AB - 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 Z × (Z/pZ) belongs to this class of groups.
ER -