Sequences of symmetry groups of infinite words

TL;DR

This paper introduces a new framework for analyzing the symmetry groups of infinite words, characterizes possible sequences of these groups, and explores their relation to periodicity and specific word classes.

Contribution

It defines the sequence of symmetry groups for infinite words, characterizes their possible structures, and relates these to properties like periodicity and specific word families.

Findings

Every subgroup of S_n can be realized as a symmetry group of some infinite word.

Sequences of symmetry groups are restricted and cannot contain certain permutations for all n.

Symmetry groups of Sturmian and Arnoux-Rauzy words are of order two for large n.

Abstract

In this paper we introduce a new notion of a sequence of symmetry groups of an infinite word. Given a subgroup $G_n$ of the symmetric group $S_n$, it acts on the set of finite words of length $n$ by permutation. We associate to an infinite word $w$ a sequence $(G_n(w))_{n\geq 1}$ of its symmetry groups: For each $n$, a symmetry group of $w$ is a subgroup $G_n(w)$ of the symmetric group $S_n$ such that $g(v)$ is a factor of $w$ for each permutation $g \in G_n(w)$ and each factor $v$ of length $n$ of $w$. We study general properties of the symmetry groups of infinite words and characterize the sequences of symmetry groups of several families of infinite words. We show that for each subgroup $G$ of $S_n$ there exists an infinite word $w$ with $G_n(w)=G$. On the other hand, the structure of possible sequences $(G_n(w))_{n\geq 1}$ is quite restrictive: we show that they cannot contain for each order $n$ certain cycles, transpositions and some other permutations. The sequences of symmetry groups can also characterize a generalized periodicity property. We prove that symmetry groups of Sturmian words and more generally Arnoux-Rauzy words are of order two for large enough $n$; on the other hand, symmetry groups of certain Toeplitz words have exponential growth.