Bounded cohomology and scl of verbal wreath products
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913860460478464 |
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| author | Bogliolo, Elena |
| author_facet | Bogliolo, Elena |
| contents | We study the bounded cohomology and the stable commutator length of verbal wreath products $Γ\wr^{_W}A$, where $A$ has trivial bounded cohomology for a sufficiently large class of coefficients.\\ We prove that the stable commutator length always vanishes, and that the bounded cohomology vanishes in positive degrees for some such verbal wreath products; including the standard restricted wreath products (extending a recent result by Monod for lamplighters groups), as well as verbal wreath products arising from n-solvable, $n$-nilpotent, and $k$-Burnside $(k = 2, 3, 4, 6)$ verbal products.\ As an application, we show that every group of type $F_p$ isometrically embeds into a group of type $F_p$ with vanishing bounded cohomology in positive degrees for a large class of coefficients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_07438 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded cohomology and scl of verbal wreath products Bogliolo, Elena Group Theory Algebraic Topology Functional Analysis 20F65, 20J05, 20E22, 46M18 We study the bounded cohomology and the stable commutator length of verbal wreath products $Γ\wr^{_W}A$, where $A$ has trivial bounded cohomology for a sufficiently large class of coefficients.\\ We prove that the stable commutator length always vanishes, and that the bounded cohomology vanishes in positive degrees for some such verbal wreath products; including the standard restricted wreath products (extending a recent result by Monod for lamplighters groups), as well as verbal wreath products arising from n-solvable, $n$-nilpotent, and $k$-Burnside $(k = 2, 3, 4, 6)$ verbal products.\ As an application, we show that every group of type $F_p$ isometrically embeds into a group of type $F_p$ with vanishing bounded cohomology in positive degrees for a large class of coefficients. |
| title | Bounded cohomology and scl of verbal wreath products |
| topic | Group Theory Algebraic Topology Functional Analysis 20F65, 20J05, 20E22, 46M18 |
| url | https://arxiv.org/abs/2501.07438 |