Lifting closed curves to finite covers of free groups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911348812677120 |
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| author | Das, Deblina Kabiraj, Arpan |
| author_facet | Das, Deblina Kabiraj, Arpan |
| contents | In this article, we show that given any integer $l\geq 2$, every closed curve $γ$ on the bouquet of $n$-circles $Γ$, admits a lift to a finite $l$-sheeted normal covering of $Γ$. Equivalently, identifying the free group $F_n$ of $n$ generators with the fundamental group of $Γ$, this statement asserts that $F_n$ is a union of $ l$-index normal subgroups for any $l\geq 2.$ The proof proceeds by explicitly constructing families of $l$-sheeted normal coverings of $Γ$, together with a characterization, in terms of necessary and sufficient conditions, of when a closed curve $γ$ on $Γ$ lifts to these covers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00481 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lifting closed curves to finite covers of free groups Das, Deblina Kabiraj, Arpan Geometric Topology M57M07 In this article, we show that given any integer $l\geq 2$, every closed curve $γ$ on the bouquet of $n$-circles $Γ$, admits a lift to a finite $l$-sheeted normal covering of $Γ$. Equivalently, identifying the free group $F_n$ of $n$ generators with the fundamental group of $Γ$, this statement asserts that $F_n$ is a union of $ l$-index normal subgroups for any $l\geq 2.$ The proof proceeds by explicitly constructing families of $l$-sheeted normal coverings of $Γ$, together with a characterization, in terms of necessary and sufficient conditions, of when a closed curve $γ$ on $Γ$ lifts to these covers. |
| title | Lifting closed curves to finite covers of free groups |
| topic | Geometric Topology M57M07 |
| url | https://arxiv.org/abs/2411.00481 |