Stable commutator length on free $\mathbb{Q}$-groups
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908683126964224 |
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| author | Fournier-Facio, Francesco |
| author_facet | Fournier-Facio, Francesco |
| contents | We study stable commutator length on free $\mathbb{Q}$-groups. We prove that every non-identity element has positive stable commutator length, and that the corresponding free group embeds isometrically. We deduce that a non-abelian free $\mathbb{Q}$-group has an infinite-dimensional space of homogeneous quasimorphisms modulo homomorphisms, answering a question of Casals-Ruiz, Garreta, and de la Nuez Gonz{á}lez. We conjecture that stable commutator length is rational on free $\mathbb{Q}$-groups. This is connected to the long-standing problem of rationality on surface groups: indeed, we show that free $\mathbb{Q}$-groups contain isometrically embedded copies of non-orientable surface groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable commutator length on free $\mathbb{Q}$-groups Fournier-Facio, Francesco Group Theory We study stable commutator length on free $\mathbb{Q}$-groups. We prove that every non-identity element has positive stable commutator length, and that the corresponding free group embeds isometrically. We deduce that a non-abelian free $\mathbb{Q}$-group has an infinite-dimensional space of homogeneous quasimorphisms modulo homomorphisms, answering a question of Casals-Ruiz, Garreta, and de la Nuez Gonz{á}lez. We conjecture that stable commutator length is rational on free $\mathbb{Q}$-groups. This is connected to the long-standing problem of rationality on surface groups: indeed, we show that free $\mathbb{Q}$-groups contain isometrically embedded copies of non-orientable surface groups. |
| title | Stable commutator length on free $\mathbb{Q}$-groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2507.14009 |