$\ell^\infty$-cohomology: amenability, relative hyperbolicity, isoperimetric inequalities and undecidability
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866915529145450496 |
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| author | Milizia, Francesco |
| author_facet | Milizia, Francesco |
| contents | We revisit Gersten's $\ell^\infty$-cohomology of groups and spaces, removing the finiteness assumptions required by the original definition while retaining its geometric nature. Mirroring the corresponding results in bounded cohomology, we provide a characterization of amenable groups using $\ell^\infty$-cohomology, and generalize Mineyev's characterization of hyperbolic groups via $\ell^\infty$-cohomology to the relative setting. We then describe how $\ell^\infty$-cohomology is related to isoperimetric inequalities. We also consider some algorithmic problems concerning $\ell^\infty$-cohomology and show that they are undecidable. In an appendix, we prove a version of the de Rham's theorem in the context of $\ell^\infty$-cohomology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_09089 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $\ell^\infty$-cohomology: amenability, relative hyperbolicity, isoperimetric inequalities and undecidability Milizia, Francesco Geometric Topology Algebraic Topology Group Theory 57M07 (Primary) 20J06, 55N25, 55N35, 58A12 (Secondary) We revisit Gersten's $\ell^\infty$-cohomology of groups and spaces, removing the finiteness assumptions required by the original definition while retaining its geometric nature. Mirroring the corresponding results in bounded cohomology, we provide a characterization of amenable groups using $\ell^\infty$-cohomology, and generalize Mineyev's characterization of hyperbolic groups via $\ell^\infty$-cohomology to the relative setting. We then describe how $\ell^\infty$-cohomology is related to isoperimetric inequalities. We also consider some algorithmic problems concerning $\ell^\infty$-cohomology and show that they are undecidable. In an appendix, we prove a version of the de Rham's theorem in the context of $\ell^\infty$-cohomology. |
| title | $\ell^\infty$-cohomology: amenability, relative hyperbolicity, isoperimetric inequalities and undecidability |
| topic | Geometric Topology Algebraic Topology Group Theory 57M07 (Primary) 20J06, 55N25, 55N35, 58A12 (Secondary) |
| url | https://arxiv.org/abs/2107.09089 |