The finite index basis property

Abstract

International audienceWe 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, namely the class of tree sets. We prove as a main result that for a uniformly recurrent tree set S, a finite bifix code X on the alphabet A is S-maximal of S-degree d if and only if it is the basis of a subgroup of index d of the free group on A

Specular Sets

“…The first one asserts that the set of return words to a given word in a recurrent specular set is a basis of a subgroup of index 2, called the even subgroup. The last one characterizes the symmetric bases of subgroups of finite index of specular groups contained in a specular set S as the finite S-maximal symmetric bifix codes contained in S. This generalizes the analogous result proved initially in [4] for Sturmian sets and extended in [8] to a more general class of sets, containing both Sturmian sets and interval exchange sets.…”

“…The following result is the counterpart for specular sets of the result holding for recurrent tree sets of characteristic 1 (see [8,Theorem 4.4]). The proof is very similar to that of Theorem 4.4 in [8] and we omit some details. Theorem 8.1 (Finite Index Basis Theorem) Let S be a recurrent specular set and let X ⊂ S be a finite symmetric bifix code.…”

Bifix codes and interval exchanges

“…This result invited us to try to extend to regular interval exchange transformations the results relating bifix codes and Sturmian words. This lead us to generalize in [4] to a large class of sets the main result of [2], namely the Finite Index Basis Theorem relating maximal bifix codes and bases of subgroups of finite index of the free group. Theorem 3.13 reveals a close connection between maximal bifix codes and interval exchange transformations.…”

Return words of linear involutions and fundamental groups

“…Indeed, in a uniformly recurrent tree set, the sets of first return words to a given word are bases of the free group on the alphabet [6]. Moreover, maximal bifix codes that are included in uniformly recurrent tree sets provide bases of subgroups of finite index of the free group [4]. Tree sets are also proved to be closed under maximal bifix decoding and under decoding with respect to return words [5].…”