Locally integrable cross sections and their intersection covolume
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912918969253888 |
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| author | Avraham-Re'em, Nachi Björklund, Michael Cullman, Rickard |
| author_facet | Avraham-Re'em, Nachi Björklund, Michael Cullman, Rickard |
| contents | We study systematically cross sections of probability preserving actions of unimodular groups and their associated transverse measures, and introduce the invariant \emph{intersection covolume} to quantify their periodicity. Our main theorem, derived from a higher order version of Kac's lemma, shows that the intersection covolume is bounded below by the intensity, with equality precisely when the action is induced by a lattice (in the sense of Mackey). We further prove that the natural cross sections of cut--and--project actions have finite intersection covolume. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20836 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Locally integrable cross sections and their intersection covolume Avraham-Re'em, Nachi Björklund, Michael Cullman, Rickard Dynamical Systems Probability 37A15 (primary), 22F10, 22D40, 28D15, 60G55, 28C10 We study systematically cross sections of probability preserving actions of unimodular groups and their associated transverse measures, and introduce the invariant \emph{intersection covolume} to quantify their periodicity. Our main theorem, derived from a higher order version of Kac's lemma, shows that the intersection covolume is bounded below by the intensity, with equality precisely when the action is induced by a lattice (in the sense of Mackey). We further prove that the natural cross sections of cut--and--project actions have finite intersection covolume. |
| title | Locally integrable cross sections and their intersection covolume |
| topic | Dynamical Systems Probability 37A15 (primary), 22F10, 22D40, 28D15, 60G55, 28C10 |
| url | https://arxiv.org/abs/2509.20836 |