Continuous cochains on Furstenberg boundaries and injectivity of the comparison map
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912632488853504 |
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| author | Bucher, Michelle Savini, Alessio |
| author_facet | Bucher, Michelle Savini, Alessio |
| contents | Monod proved that any continuous cohomology of a semisimple Lie group $G$ can be represented by a measurable cocycle on the associated Furstenberg boundary, which we upgraded to an alternating cocycle. In the current paper we improve that result by showing that we can actually take a representing cocycle which is continuous on an explicit subset of generic tuples. We give an analogous result in the case of bounded cohomology. Finally, we exploit this characterization to prove the injectivity of the comparison map in degree $3$ for $\mathrm{Isom}^\circ(\mathbb{H}_{\mathbb{C}}^n)$, when $n \geq 2$, and in degree $4$ for $\mathrm{Isom}^\circ(\mathbb{H}^n_{\mathbb{R}})$, when $n \geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05333 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuous cochains on Furstenberg boundaries and injectivity of the comparison map Bucher, Michelle Savini, Alessio Group Theory 22E41, 57T10 Monod proved that any continuous cohomology of a semisimple Lie group $G$ can be represented by a measurable cocycle on the associated Furstenberg boundary, which we upgraded to an alternating cocycle. In the current paper we improve that result by showing that we can actually take a representing cocycle which is continuous on an explicit subset of generic tuples. We give an analogous result in the case of bounded cohomology. Finally, we exploit this characterization to prove the injectivity of the comparison map in degree $3$ for $\mathrm{Isom}^\circ(\mathbb{H}_{\mathbb{C}}^n)$, when $n \geq 2$, and in degree $4$ for $\mathrm{Isom}^\circ(\mathbb{H}^n_{\mathbb{R}})$, when $n \geq 2$. |
| title | Continuous cochains on Furstenberg boundaries and injectivity of the comparison map |
| topic | Group Theory 22E41, 57T10 |
| url | https://arxiv.org/abs/2510.05333 |