The stable Albanese homology of the IA-automorphism groups of free groups
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910421662826496 |
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| author | Katada, Mai |
| author_facet | Katada, Mai |
| contents | The IA-automorphism group $\operatorname{IA}_n$ of the free group $F_n$ of rank $n$ is a normal subgroup of the automorphism group $\operatorname{Aut}(F_n)$ of $F_n$. We study the Albanese homology of $\operatorname{IA}_n$, which is the quotient of the rational homology of $\operatorname{IA}_n$ defined as the image of the map induced by the abelianization map of $\operatorname{IA}_n$ on homology. The Albanese homology of $\operatorname{IA}_n$ is an algebraic $\operatorname{GL}(n,\mathbb{Q})$-representation. We determine the representation structure of the Albanese homology of $\operatorname{IA}_n$ for $n$ greater than or equal to three times the homological degree. We also determine the structure of the stable Albanese homology of the analogue of $\operatorname{IA}_n$ to the outer automorphism group of $F_n$. Moreover, we identify the relation between the stable Albanese (co)homology of $\operatorname{IA}_n$ and the stable cohomology of $\operatorname{Aut}(F_n)$ with certain twisted coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15901 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The stable Albanese homology of the IA-automorphism groups of free groups Katada, Mai Algebraic Topology Group Theory 20F28, 20J06 The IA-automorphism group $\operatorname{IA}_n$ of the free group $F_n$ of rank $n$ is a normal subgroup of the automorphism group $\operatorname{Aut}(F_n)$ of $F_n$. We study the Albanese homology of $\operatorname{IA}_n$, which is the quotient of the rational homology of $\operatorname{IA}_n$ defined as the image of the map induced by the abelianization map of $\operatorname{IA}_n$ on homology. The Albanese homology of $\operatorname{IA}_n$ is an algebraic $\operatorname{GL}(n,\mathbb{Q})$-representation. We determine the representation structure of the Albanese homology of $\operatorname{IA}_n$ for $n$ greater than or equal to three times the homological degree. We also determine the structure of the stable Albanese homology of the analogue of $\operatorname{IA}_n$ to the outer automorphism group of $F_n$. Moreover, we identify the relation between the stable Albanese (co)homology of $\operatorname{IA}_n$ and the stable cohomology of $\operatorname{Aut}(F_n)$ with certain twisted coefficients. |
| title | The stable Albanese homology of the IA-automorphism groups of free groups |
| topic | Algebraic Topology Group Theory 20F28, 20J06 |
| url | https://arxiv.org/abs/2404.15901 |