On the Hausdorff spectra of free pro-$p$ groups and certain $p$-adic analytic groups
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908374328672256 |
|---|---|
| author | Heras, Iker de las Klopsch, Benjamin Thillaisundaram, Anitha |
| author_facet | Heras, Iker de las Klopsch, Benjamin Thillaisundaram, Anitha |
| contents | We establish that finitely generated non-abelian direct products $G$ of free pro-$p$ groups have full Hausdorff spectrum with respect to the lower $p$-series $\mathcal{L}$. This complements similar results with respect to other standard filtration series and a recent theorem showing that the Hausdorff spectrum $\text{hspec}^\mathcal{L}(G)$ of a $p$-adic analytic pro-$p$ group $G$ is discrete and consists of at most $2^{\dim(G)}$ rational numbers.
The latter also left some room for improvement regarding the upper bound. Indeed, for finitely generated nilpotent pro-$p$ groups $G$ we obtain the stronger assertion that the cardinality of the Hausdorff spectrum is at most the analytic dimension of $G$. Moreover, we produce a corresponding result when the $p$-adic analytic pro-$p$ group $G$ is just infinite, which holds not just for the lower $p$-series but for arbitrary filtration series.
Finally, we show that, if $G$ is a countably based pro-$p$ group with an open subgroup mapping onto the free abelian pro-$p$ group $\mathbb{Z}_p \oplus \mathbb{Z}_p$, then for every prescribed finite set $\{0,1\} \subseteq X \subseteq [0,1]$ there is a filtration series $\mathcal{S}$ such that $\text{hspec}^\mathcal{S}(G) = X$; in particular, $|\text{hspec}^{\mathcal{S}}(G)|$ is unbounded, as $\mathcal{S}$ runs through all filtration series of $G$ with $|\text{hspec}^{\mathcal{S}}(G)| < \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Hausdorff spectra of free pro-$p$ groups and certain $p$-adic analytic groups Heras, Iker de las Klopsch, Benjamin Thillaisundaram, Anitha Group Theory We establish that finitely generated non-abelian direct products $G$ of free pro-$p$ groups have full Hausdorff spectrum with respect to the lower $p$-series $\mathcal{L}$. This complements similar results with respect to other standard filtration series and a recent theorem showing that the Hausdorff spectrum $\text{hspec}^\mathcal{L}(G)$ of a $p$-adic analytic pro-$p$ group $G$ is discrete and consists of at most $2^{\dim(G)}$ rational numbers. The latter also left some room for improvement regarding the upper bound. Indeed, for finitely generated nilpotent pro-$p$ groups $G$ we obtain the stronger assertion that the cardinality of the Hausdorff spectrum is at most the analytic dimension of $G$. Moreover, we produce a corresponding result when the $p$-adic analytic pro-$p$ group $G$ is just infinite, which holds not just for the lower $p$-series but for arbitrary filtration series. Finally, we show that, if $G$ is a countably based pro-$p$ group with an open subgroup mapping onto the free abelian pro-$p$ group $\mathbb{Z}_p \oplus \mathbb{Z}_p$, then for every prescribed finite set $\{0,1\} \subseteq X \subseteq [0,1]$ there is a filtration series $\mathcal{S}$ such that $\text{hspec}^\mathcal{S}(G) = X$; in particular, $|\text{hspec}^{\mathcal{S}}(G)|$ is unbounded, as $\mathcal{S}$ runs through all filtration series of $G$ with $|\text{hspec}^{\mathcal{S}}(G)| < \infty$. |
| title | On the Hausdorff spectra of free pro-$p$ groups and certain $p$-adic analytic groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2505.16417 |