Self-similarity of the classical $p$-adic Lie groups and Lie algebras

TL;DR

This paper demonstrates that certain classical p-adic Lie groups can act self-similarly on regular rooted trees, revealing new structural properties linked to ramification indices and Lie lattice endomorphisms.

Contribution

It introduces a method to identify ramification indices allowing faithful self-similar actions of classical p-adic Lie groups on rooted trees, expanding understanding of their algebraic and geometric structures.

Findings

Identified infinite lists of ramification indices for self-similar actions.

Connected self-similarity of p-adic Lie groups with virtual endomorphisms of Lie lattices.

Computed indices of principal congruence subgroups for various local rings.

Abstract

We exhibit infinite lists of ramification indices $\delta$ for which the classical Lie groups over the ring of integers of $p$-adic fields admit a faithful self-similar action on a regular rooted $\delta$-ary tree in such a way that the action is transitive on the first level. These results follow from the study of virtual endomorphisms of the classical Lie lattices over the same type of rings. In order to compute the ramification indices for all the types of groups treated in the paper, we compute the indices of principal congruence subgroups of the orthogonal groups for a class of local rings.