Arithmetic aspects of self-similar groups

TL;DR

This paper establishes a deep connection between the arithmetic nature of certain lattices in semisimple groups and their ability to act self-similarly on finite-valency rooted trees, revealing new structural insights.

Contribution

It proves that irreducible lattices are arithmetic if and only if they have a faithful self-similar action on a finite-valency rooted tree, linking algebraic and dynamical properties.

Findings

Irreducible lattices are arithmetic iff they admit faithful self-similar actions.

Self-similar actions characterize the arithmeticity of lattices.

Provides a criterion for identifying arithmetic lattices via tree actions.

Abstract

We prove that an irreducible lattice in a semisimple algebraic group is virtually isomorphic to an arithmetic lattice if and only if it admits a faithful self-similar action on a rooted tree of finite valency.