Bounded cohomology, quotient extensions, and hierarchical hyperbolicity
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917186968223744 |
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| author | Fournier-Facio, Francesco Mangioni, Giorgio Sisto, Alessandro |
| author_facet | Fournier-Facio, Francesco Mangioni, Giorgio Sisto, Alessandro |
| contents | We call a central extension bounded if its Euler class is represented by a bounded cocycle. We prove that a bounded central extension of a hierarchically hyperbolic group (HHG) is still a HHG; conversely if a central extension is a HHG, then the extension is bounded, and under a further mild assumption the quotient is commensurable to a HHG. Motivated by questions on hierarchical hyperbolicity of quotients of mapping class groups, we therefore consider the general problem of determining when a quotient of a bounded central extension is still bounded, which we prove to be equivalent to an extendability problem for quasihomomorphisms. Finally, we show that quotients of the 4-strands braid group by suitable powers of a pseudo-Anosov are HHG, and in fact bounded central extensions of some HHG. We also speculate on how to extend the previous result to all mapping class groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20462 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded cohomology, quotient extensions, and hierarchical hyperbolicity Fournier-Facio, Francesco Mangioni, Giorgio Sisto, Alessandro Group Theory 20F65 (Primary), 57K20, 20J06 (Secondary) We call a central extension bounded if its Euler class is represented by a bounded cocycle. We prove that a bounded central extension of a hierarchically hyperbolic group (HHG) is still a HHG; conversely if a central extension is a HHG, then the extension is bounded, and under a further mild assumption the quotient is commensurable to a HHG. Motivated by questions on hierarchical hyperbolicity of quotients of mapping class groups, we therefore consider the general problem of determining when a quotient of a bounded central extension is still bounded, which we prove to be equivalent to an extendability problem for quasihomomorphisms. Finally, we show that quotients of the 4-strands braid group by suitable powers of a pseudo-Anosov are HHG, and in fact bounded central extensions of some HHG. We also speculate on how to extend the previous result to all mapping class groups. |
| title | Bounded cohomology, quotient extensions, and hierarchical hyperbolicity |
| topic | Group Theory 20F65 (Primary), 57K20, 20J06 (Secondary) |
| url | https://arxiv.org/abs/2505.20462 |