On the $\ell^2$-Betti numbers and algebraic fibring of the (outer) automorphism group of a right-angled Artin group
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917103146106880 |
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| author | Ferrer, Marcos Escartín |
| author_facet | Ferrer, Marcos Escartín |
| contents | We compute the first $\ell^2$-Betti number of the automorphism and outer automorphism groups of arbitrary right-angled Artin groups (RAAGs), providing a complete characterization of when it is non-zero. We also analyse the algebraic fibring of the pure symmetric automorphism groups $\mathrm{PSA}(A_Γ)$ and $\mathrm{PSO}(A_Γ)$ and the virtual algebraic fibring of $\mathrm{Out}(A_Γ)$ in the case when $A_Γ$ admits no non-inner partial conjugation. In the transvection-free case, we show that $β_1^{(2)}(\mathrm{Out}(A_Γ)) = 0$ if and only if $\mathrm{Out}(A_Γ)$ virtually fibres. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06587 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $\ell^2$-Betti numbers and algebraic fibring of the (outer) automorphism group of a right-angled Artin group Ferrer, Marcos Escartín Group Theory We compute the first $\ell^2$-Betti number of the automorphism and outer automorphism groups of arbitrary right-angled Artin groups (RAAGs), providing a complete characterization of when it is non-zero. We also analyse the algebraic fibring of the pure symmetric automorphism groups $\mathrm{PSA}(A_Γ)$ and $\mathrm{PSO}(A_Γ)$ and the virtual algebraic fibring of $\mathrm{Out}(A_Γ)$ in the case when $A_Γ$ admits no non-inner partial conjugation. In the transvection-free case, we show that $β_1^{(2)}(\mathrm{Out}(A_Γ)) = 0$ if and only if $\mathrm{Out}(A_Γ)$ virtually fibres. |
| title | On the $\ell^2$-Betti numbers and algebraic fibring of the (outer) automorphism group of a right-angled Artin group |
| topic | Group Theory |
| url | https://arxiv.org/abs/2509.06587 |