The lower $p$-series of analytic pro-$p$ groups and Hausdorff dimension
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911146085187584 |
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| author | Heras, Iker de las Klopsch, Benjamin Thillaisundaram, Anitha |
| author_facet | Heras, Iker de las Klopsch, Benjamin Thillaisundaram, Anitha |
| contents | Let $G$ be a $p$-adic analytic pro-$p$ group of dimension $d$. We produce an approximate series which descends regularly in strata and whose terms deviate from the lower $p$-series in a uniformly bounded way. This brings to light a new set of rational invariants, canonically associated to $G$, that yield the aforementioned uniform bound and that restrict the possible values for the Hausdorff dimensions of closed subgroups of $G$ with respect to the lower $p$-series. In particular, the Hausdorff spectrum of $G$ with respect to the lower $p$-series is discrete and consists of at most $2^d$ rational numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06876 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The lower $p$-series of analytic pro-$p$ groups and Hausdorff dimension Heras, Iker de las Klopsch, Benjamin Thillaisundaram, Anitha Group Theory Let $G$ be a $p$-adic analytic pro-$p$ group of dimension $d$. We produce an approximate series which descends regularly in strata and whose terms deviate from the lower $p$-series in a uniformly bounded way. This brings to light a new set of rational invariants, canonically associated to $G$, that yield the aforementioned uniform bound and that restrict the possible values for the Hausdorff dimensions of closed subgroups of $G$ with respect to the lower $p$-series. In particular, the Hausdorff spectrum of $G$ with respect to the lower $p$-series is discrete and consists of at most $2^d$ rational numbers. |
| title | The lower $p$-series of analytic pro-$p$ groups and Hausdorff dimension |
| topic | Group Theory |
| url | https://arxiv.org/abs/2402.06876 |