Self-similar groups, automatic sequences, and unitriangular representations
Group Theory
2014-09-18 v1
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 -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.
Cite
@article{arxiv.1409.5027,
title = {Self-similar groups, automatic sequences, and unitriangular representations},
author = {R. Grigorchuk and Y. Leonov and V. Nekrashevych and V. Sushchansky},
journal= {arXiv preprint arXiv:1409.5027},
year = {2014}
}
Comments
56 pages, 7 figures