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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.26158 |
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| _version_ | 1866908916871331840 |
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| author | Izeki, Hiroyasu Ji, Ran Karlsson, Anders Wu, Yunhui |
| author_facet | Izeki, Hiroyasu Ji, Ran Karlsson, Anders Wu, Yunhui |
| contents | We prove that finitely generated amenable groups acting on CAT(0) spaces satisfy the following alternative: either every action on a geodesically complete CAT(0) space with bounded geometry (or finite dimension) has a global fixed point, or the group admits a fixed-point-free action on $\mathbb{R}^n$. As a consequence, finitely generated amenable torsion groups and finitely generated virtually simple amenable groups cannot act nontrivially on geodesically complete CAT(0) spaces with bounded geometry or on finite-dimensional complete CAT(0) spaces.
The proof relies on a Kazhdan-type rigidity theorem for groups with the Euclidean fixed point property: if such a group acts on a geodesically complete CAT(0) space of bounded geometry with almost fixed points, then it has a genuine fixed point. This yields several further corollaries, including a rigidity dichotomy for drift and that any finitely generated torsion group acting on a geodesically complete visibility CAT(0) space with bounded geometry must have a global fixed point. These results make substantial progress on the longstanding problem of understanding actions of torsion groups on CAT(0) spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26158 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A CAT(0) alternative for amenable groups and a Kazhdan-type rigidity principle Izeki, Hiroyasu Ji, Ran Karlsson, Anders Wu, Yunhui Group Theory Metric Geometry 20F65, 22D55, 51K10, 53C24, 20K10 (Primary) 60B15, 60G50 (Secondary) We prove that finitely generated amenable groups acting on CAT(0) spaces satisfy the following alternative: either every action on a geodesically complete CAT(0) space with bounded geometry (or finite dimension) has a global fixed point, or the group admits a fixed-point-free action on $\mathbb{R}^n$. As a consequence, finitely generated amenable torsion groups and finitely generated virtually simple amenable groups cannot act nontrivially on geodesically complete CAT(0) spaces with bounded geometry or on finite-dimensional complete CAT(0) spaces. The proof relies on a Kazhdan-type rigidity theorem for groups with the Euclidean fixed point property: if such a group acts on a geodesically complete CAT(0) space of bounded geometry with almost fixed points, then it has a genuine fixed point. This yields several further corollaries, including a rigidity dichotomy for drift and that any finitely generated torsion group acting on a geodesically complete visibility CAT(0) space with bounded geometry must have a global fixed point. These results make substantial progress on the longstanding problem of understanding actions of torsion groups on CAT(0) spaces. |
| title | A CAT(0) alternative for amenable groups and a Kazhdan-type rigidity principle |
| topic | Group Theory Metric Geometry 20F65, 22D55, 51K10, 53C24, 20K10 (Primary) 60B15, 60G50 (Secondary) |
| url | https://arxiv.org/abs/2603.26158 |