On self-similarity of $p$-adic analytic pro-$p$ groups of small dimension
| Autor: | Noseda, Francesco, Snopce, Ilir |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Journal of Algebra 540 (2019) 317-345 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.jalgebra.2019.09.003 |
| Popis: | Given a torsion-free $p$-adic analytic pro-$p$ group $G$ with $\mathrm{dim}(G) < p$, we show that the self-similar actions of $G$ on regular rooted trees can be studied through the virtual endomorphisms of the associated $\mathbb{Z}_p$-Lie lattice. We explicitly classify 3-dimensional unsolvable $\mathbb{Z}_p$-Lie lattices for $p$ odd, and study their virtual endomorphisms. Together with Lazard's correspondence, this allows us to classify 3-dimensional unsolvable torsion-free $p$-adic analytic pro-$p$ groups for $p\geqslant 5$, and to determine which of them admit a faithful self-similar action on a $p$-ary tree. In particular, we show that no open subgroup of $SL_1^1(\Delta_p)$ admits such an action. On the other hand, we prove that all the open subgroups of $SL_2^{\triangle}(\mathbb{Z}_p)$ admit faithful self-similar actions on regular rooted trees. Comment: Corrections to the statements of Conjecture 2.9, Conjecture 2.10, Lemma 2.11, and Lemma 2.12, and to the proof of Theorem D |
| Databáze: | arXiv |
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