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| Main Author: | |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2102.11458 |
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Table of 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.