تقرير
Liftable self-similar groups and scale groups
| العنوان: | Liftable self-similar groups and scale groups |
|---|---|
| المؤلفون: | Grigorchuk, Rostislav, Savchuk, Dmytro |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Group Theory, 20E08, 22D05 |
| الوصف: | We canonically identify the groups of isometries and dilations of local fields and their rings of integers with subgroups of the automorphism group of the $(d+1)$-regular tree $\widetilde T_{d+1}$, where $d$ is the residual degree. Then we introduce the class of liftable self-similar groups acting on a $d$-regular rooted tree whose ascending HNN extensions act faithfully and vertex transitively on $\widetilde T_{d+1}$ fixing one of the ends. The closures of these extensions in $\mathrm{Aut}(\widetilde T_{d+1})$ are totally disconnected locally compact group that belong to the class of scale groups. We give numerous examples of liftable groups coming from self-similar groups acting essentially freely or groups admitting finite $L$-presentations. In particular, we show that the finitely presented group constructed by the first author and the finitely presented HNN extension of the Basilica group embed into the group $\mathcal D(\mathbb Q_2)$ of dilations of the field $\mathbb Q_2$ of $2$-adic numbers. These actions, translated to $\widetilde T_3$, are 2-transitive on the punctured boundary of $\widetilde T_3$. Also we explore scale-invariant groups with the purpose of getting new examples of scale groups. Comment: 49 pages, 9 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2312.05427 |
| رقم الأكسشن: | edsarx.2312.05427 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |