Liftable self-similar groups and scale groups - Texas A&M University (TAMU) Scholar

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.

Liftable self-similar groups and scale groups Institutional Repository Document