The Kervaire conjecture and the minimal complexity of surfaces
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866908619261345792 |
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| author | Chen, Lvzhou |
| author_facet | Chen, Lvzhou |
| contents | The Kervaire conjecture asserts that adding a generator and then a relator to a nontrivial group always results in a nontrivial group. We introduce new methods from stable commutator length to study this type of problems about nontriviality of one-relator quotients. Roughly, we show that surfaces in certain HNN extensions bounding a given word have complexity no less than the complexity of its boundary. A consequence of this is a Freiheitssatz theorem for HNN extensions, which in particular implies and gives a new proof of Klyachko's theorem that confirms the Kervaire conjecture for torsion-free groups. As another application, we also generalize the following theorem of Klyachko-Lurye to HNN extensions: For any group $G$ and the quotient $Q$ of $G\star\mathbb{Z}$ by any proper power $w^m$ with $w\in G\star\mathbb{Z}$ projecting to $1\in\mathbb{Z}$, the natural map $G\to Q$ is injective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_09811 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Kervaire conjecture and the minimal complexity of surfaces Chen, Lvzhou Group Theory Geometric Topology The Kervaire conjecture asserts that adding a generator and then a relator to a nontrivial group always results in a nontrivial group. We introduce new methods from stable commutator length to study this type of problems about nontriviality of one-relator quotients. Roughly, we show that surfaces in certain HNN extensions bounding a given word have complexity no less than the complexity of its boundary. A consequence of this is a Freiheitssatz theorem for HNN extensions, which in particular implies and gives a new proof of Klyachko's theorem that confirms the Kervaire conjecture for torsion-free groups. As another application, we also generalize the following theorem of Klyachko-Lurye to HNN extensions: For any group $G$ and the quotient $Q$ of $G\star\mathbb{Z}$ by any proper power $w^m$ with $w\in G\star\mathbb{Z}$ projecting to $1\in\mathbb{Z}$, the natural map $G\to Q$ is injective. |
| title | The Kervaire conjecture and the minimal complexity of surfaces |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2302.09811 |