Liftability and Contracting Property of Multi-EGS Groups

TL;DR

The paper addresses liftability and contraction properties of multi-EGS groups acting on $p$-ary rooted trees, deriving explicit liftings under a column-condition on the defining datum and showing the resulting ascending HNN extensions embed into $\mathrm{Aut}(\widetilde{T}_{p+1})$ with closures that are scale groups. The main result provides a concrete lifting $\sigma$ when there exist $m,k,j$ with $e_{k,j}^{(m)}\neq 0$ and $e_{i,p-j}^{(l)}=0$, and it yields a precise abelianization $G_{\mathbf E}/G_{\mathbf E}'\cong(\mathbb Z/p\mathbb Z)^{1+r_1+\dots+r_p}$. The authors also compute the contracting nucleus $\mathcal N_{\mathbf E}=\langle a\rangle \cup \bigcup_{l=1}^p \langle b_i^{(l)}\rangle$ with $|\mathcal N_{\mathbf E}|=\sum_{l=1}^p p^{r_l}$, establishing contraction via Nekrashevych's framework and enabling potential word-problem and cryptographic applications. They specialize the general theorem to multi-edge spinal groups and to EGS-groups, and discuss how these liftable and contracting structures fit into Willis' scale-group theory.

Abstract

We provide sufficient conditions for the multi-EGS groups to be liftable and thus produce new examples of groups acting transitively on regular trees of finite degree stabilizing one of the ends, whose closures are scale groups as defined by Willis. Additionally, we explicitly compute the contracting nuclei of the groups in this class. We also specialize our results to the classes of multi-edge spinal group and EGS-groups.

Theorems & Definitions (16)

• Theorem A
• Corollary B
• Theorem C
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• proof
• Definition 3.2
• Corollary 3.3