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| Format: | Preprint |
| Published: |
2021
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| Online Access: | https://arxiv.org/abs/2102.11458 |
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| _version_ | 1866915455110742016 |
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| author | Costa, Iván Sadofschi |
| author_facet | Costa, Iván Sadofschi |
| contents | In this series of two articles, we prove that every action of a finite group $G$ on a finite and contractible $2$-complex has a fixed point. The proof goes by constructing a nontrivial representation of the fundamental group of each of the acyclic $2$-dimensional $G$-complexes constructed by Oliver and Segev. In the first part we develop the necessary theory and cover the cases where $G=\mathrm{PSL}_2(2^n)$, $G=\mathrm{PSL}_2(q)$ with $q\equiv 3\pmod 8$ or $G=\mathrm{Sz}(2^n)$. The cases $G=\mathrm{PSL}_2(q)$ with $q\equiv 5\pmod 8$ are addressed in the second part. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_11458 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Group actions on contractible $2$-complexes I Costa, Iván Sadofschi Algebraic Topology Group Theory Representation Theory 57M60 In this series of two articles, we prove that every action of a finite group $G$ on a finite and contractible $2$-complex has a fixed point. The proof goes by constructing a nontrivial representation of the fundamental group of each of the acyclic $2$-dimensional $G$-complexes constructed by Oliver and Segev. In the first part we develop the necessary theory and cover the cases where $G=\mathrm{PSL}_2(2^n)$, $G=\mathrm{PSL}_2(q)$ with $q\equiv 3\pmod 8$ or $G=\mathrm{Sz}(2^n)$. The cases $G=\mathrm{PSL}_2(q)$ with $q\equiv 5\pmod 8$ are addressed in the second part. |
| title | Group actions on contractible $2$-complexes I |
| topic | Algebraic Topology Group Theory Representation Theory 57M60 |
| url | https://arxiv.org/abs/2102.11458 |