Torsion groups of subexponential growth cannot act on finite-dimensional CAT(0)-spaces without a fixed point
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| Format: | Preprint |
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2024
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| _version_ | 1866914896930668544 |
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| author | Izeki, Hiroyasu Karlsson, Anders |
| author_facet | Izeki, Hiroyasu Karlsson, Anders |
| contents | We show that finitely generated groups which are Liouville and without infinite finite-dimensional linear representations must have a global fixed point whenever they act by isometry on a finite-dimensional complete CAT(0)-space. This provides a partial answer to an old question in geometric group theory and proves partly a conjecture formulated by Norin, Osajda, and Przytycki. It applies in particular to Grigorchuk's groups of intermediate growth and other branch groups as well as to simple groups with the Liouville property such as those found by Matte Bon and by Nekrashevych. The method of proof uses ultralimits, equivariant harmonic maps, subharmonic functions, horofunctions and random walks. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_19273 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torsion groups of subexponential growth cannot act on finite-dimensional CAT(0)-spaces without a fixed point Izeki, Hiroyasu Karlsson, Anders Group Theory 20F65, 20K10, 58E20 We show that finitely generated groups which are Liouville and without infinite finite-dimensional linear representations must have a global fixed point whenever they act by isometry on a finite-dimensional complete CAT(0)-space. This provides a partial answer to an old question in geometric group theory and proves partly a conjecture formulated by Norin, Osajda, and Przytycki. It applies in particular to Grigorchuk's groups of intermediate growth and other branch groups as well as to simple groups with the Liouville property such as those found by Matte Bon and by Nekrashevych. The method of proof uses ultralimits, equivariant harmonic maps, subharmonic functions, horofunctions and random walks. |
| title | Torsion groups of subexponential growth cannot act on finite-dimensional CAT(0)-spaces without a fixed point |
| topic | Group Theory 20F65, 20K10, 58E20 |
| url | https://arxiv.org/abs/2404.19273 |