Self-similar groups, automatic sequences, and unitriangular representations

TL;DR

This paper explores the connection between self-similar groups, automatic sequences, and linear representations, revealing that automaton-generated groups produce automatic matrices and that p-adic actions lead to infinite triangular matrix representations.

Contribution

It establishes a novel link between automaton-generated groups and automatic sequences, and characterizes linear representations of p-adic automorphism groups as infinite triangular matrices.

Findings

Automaton-generated groups produce automatic matrices.

p-adic automorphisms correspond to infinite triangular matrix representations.

The work relates automorphism height to matrix structure.

Abstract

We study natural linear representations of self-similar groups over finite fields. In particular, we show that if the group is generated by a finite automaton, then obtained matrices are automatic. This shows a new relation between two separate notions of automaticity: groups generated by automata and automatic sequences. We also show that if the group acts on the tree by $p$-adic automorphisms, then the corresponding linear representation is a representation by infinite triangular matrices. We relate this observation with the notion of height of an automorphism of a rooted tree due to L.Kaloujnine.