Combinatorics of the subshift associated to Grigorchuk's group

TL;DR

This paper investigates the combinatorial structure of a subshift linked to Grigorchuk's group, revealing its applications in embedding the group into topological full groups and analyzing spectral properties of associated Laplacians.

Contribution

It provides a detailed combinatorial analysis of the subshift from Grigorchuk's group, connecting it to group embeddings and spectral theory.

Findings

Characterization of the subshift's combinatorial properties

Application to embedding Grigorchuk's group into topological full groups

Insights into spectral properties of Laplacians on Schreier graphs

Abstract

We study combinatorial properties of the subshift induced by the substitution that describes Lysenok's presentation of Grigorchuk's group of intermediate growth by generators and relators. This subshift has recently appeared in two different contexts: on one hand, it allowed to embed Grigorchuk's group in a topological full group, and on the other hand, it was useful in the spectral theory of Laplacians on the associated Schreier graphs.