Liftable self-similar groups and scale groups

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 ) (d+1) -regular tree T ~ d + 1 Td+1 , where d d is the residual degree. Then we introduce the class of liftable self-similar groups acting on a d d -regular rooted tree whose ascending HNN extensions act faithfully and vertex transitively on T ~ d + 1 Td+1 fixing one of the ends. The closures of these extensions in A u t ( T ~ d + 1 ) Aut( Td+1) are totally disconnected locally compact groups that belong to the class of scale groups as defined by Willis [Scale groups, 2022]. We give numerous examples of liftable groups coming from self-similar groups acting essentially freely on the boundaries of rooted trees or groups admitting finite L L -presentations. In particular, we show that the finitely presented group constructed by the first author [Mat. Sb. 189 (1998), pp. 79–100] and the finitely presented HNN extension of the Basilica group constructed by Grigorchuk and Żuk [Spectral properties of a torsion-free weakly branch group defined by a three state automaton, Amer. Math. Soc., Providence, RI, 2002, pp. 57–82] embed into the group D ( Q 2 ) D(Q_2) of dilations of the field Q 2 Q_2 of 2 2 -adic numbers. These actions, translated to T ~ 3 T_3 , are 2-transitive on the punctured boundary of T ~ 3 T_3 . Also we explore scale-invariant groups studied by Nekrashevych and Pete [Groups Geom. Dyn. 5 (2011), pp. 139–167] with the purpose of getting new examples of scale groups.