On the stability of hyperbolicity under quantitative measure equivalence
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908377262587904 |
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| author | Delabie, Thiebout Koivisto, Juhani Maître, François Le Tessera, Romain |
| author_facet | Delabie, Thiebout Koivisto, Juhani Maître, François Le Tessera, Romain |
| contents | A well-known result of Shalom says that lattices in SO$(n,1)$ are $\mathrm{L}^p$ measure equivalent for all $p<n-1$. His proof actually yields the following stronger statement: the natural coupling resulting from a suitable choice of fundamental domains from a uniform lattice to a non-uniform one is $(\mathrm{L}^p,\mathrm{L}^{\infty})$. Moreover, it is easy to see that the coupling is cobounded: the fundamental domain of the uniform lattice is contained in a union of finitely many translates of the fundamental domain of the non-uniform one. The purpose of this note is to prove that this statement is sharp in the following sense: if a ME-coupling from a hyperbolic group to a non-hyperbolic group is cobounded and $(\mathrm{L}^p,\mathrm{L}^{\infty})$, then $p$ must be less than some $p_0$ only depending on the hyperbolic group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the stability of hyperbolicity under quantitative measure equivalence Delabie, Thiebout Koivisto, Juhani Maître, François Le Tessera, Romain Group Theory Metric Geometry A well-known result of Shalom says that lattices in SO$(n,1)$ are $\mathrm{L}^p$ measure equivalent for all $p<n-1$. His proof actually yields the following stronger statement: the natural coupling resulting from a suitable choice of fundamental domains from a uniform lattice to a non-uniform one is $(\mathrm{L}^p,\mathrm{L}^{\infty})$. Moreover, it is easy to see that the coupling is cobounded: the fundamental domain of the uniform lattice is contained in a union of finitely many translates of the fundamental domain of the non-uniform one. The purpose of this note is to prove that this statement is sharp in the following sense: if a ME-coupling from a hyperbolic group to a non-hyperbolic group is cobounded and $(\mathrm{L}^p,\mathrm{L}^{\infty})$, then $p$ must be less than some $p_0$ only depending on the hyperbolic group. |
| title | On the stability of hyperbolicity under quantitative measure equivalence |
| topic | Group Theory Metric Geometry |
| url | https://arxiv.org/abs/2411.10250 |