The finite index basis property

TL;DR

This paper establishes a deep connection between bifix codes, symbolic dynamical systems, and free groups, specifically characterizing certain bifix codes as bases of subgroups of free groups with finite index.

Contribution

It introduces the class of tree sets within symbolic dynamics and proves that $F$-maximal bifix codes correspond exactly to bases of subgroups of finite index in free groups.

Findings

Characterization of $F$-maximal bifix codes as free group bases.

Introduction of tree sets as a class of factor sets with linear complexity.

Extension of known results from Sturmian words to more general symbolic systems.

Abstract

We describe in this paper a connection between bifix codes, symbolic dynamical systems and free groups. This is in the spirit of the connection established previously for the symbolic systems corresponding to Sturmian words. We introduce a class of sets of factors of an infinite word with linear factor complexity containing Sturmian sets and regular interval exchange sets, namemly the class of tree sets. We prove as a main result that for a uniformly recurrent tree set $F$, a finite bifix code $X$ on the alphabet $A$ is $F$-maximal of $F$-degree $d$ if and only if it is the basis of a subgroup of index $d$ of the free group on $A$.