Measure equivalence and sofic approximations
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911025891115008 |
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| author | Delabie, Thiebout Koivisto, Juhani Tessera, Romain |
| author_facet | Delabie, Thiebout Koivisto, Juhani Tessera, Romain |
| contents | We introduce a technique for producing a measure coupling between two sofic groups from a family of maps between their sofic approximations. We exploit this to construct measure couplings between pairs of groups with prescribed integrability conditions. As an application we show that solvable Baumslag-Solitar groups, Lamplighters and the group SOL are all exponentially measure equivalent to one another: in particular they are L^p measure equivalent for all p. This is in sharp contrast with the fact that these groups are in general not quasi-isometric to one another: indeed, for instance the lamplighter with lamp group Z/3Z is not quasi-isometric the lamplighter with lamp group Z/2Z. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21750 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Measure equivalence and sofic approximations Delabie, Thiebout Koivisto, Juhani Tessera, Romain Group Theory Metric Geometry 20F65, 37A20, 28D15, 20F18, 22E25, 51F30 We introduce a technique for producing a measure coupling between two sofic groups from a family of maps between their sofic approximations. We exploit this to construct measure couplings between pairs of groups with prescribed integrability conditions. As an application we show that solvable Baumslag-Solitar groups, Lamplighters and the group SOL are all exponentially measure equivalent to one another: in particular they are L^p measure equivalent for all p. This is in sharp contrast with the fact that these groups are in general not quasi-isometric to one another: indeed, for instance the lamplighter with lamp group Z/3Z is not quasi-isometric the lamplighter with lamp group Z/2Z. |
| title | Measure equivalence and sofic approximations |
| topic | Group Theory Metric Geometry 20F65, 37A20, 28D15, 20F18, 22E25, 51F30 |
| url | https://arxiv.org/abs/2506.21750 |