Return words of linear involutions and fundamental groups

Abstract

We investigate the natural codings of linear involutions. We deduce from the geometric representation of linear involutions as Poincar\'e maps of measured foliations a suitable definition of return words which yields that the set of first return words to a given word is a symmetric basis of the free group on the underlying alphabet $A$. The set of first return words with respect to a subgroup of finite index $G$ of the free group on $A$ is also proved to be a symmetric basis of $G$

Neutral and tree sets of arbitrary characteristic

“…More precisely, we prove that the natural coding of an interval exchange transformation without connections of length ≥ 1 is a tree set and that the natural coding of a linear involution without connections is a tree set of characteristic 2. This extends a result in [5] concerning interval exchange without connections as well as a result of [9] concerning linear involutions without connection. Acknowledgement..…”

Enumeration Formulæ in Neutral Sets

“…Interval exchange transformations without connections, also called regular interval exchange transformations, are well studied (see, for example, [13] and [6]). The natural coding of a linear involutions without connection (see [9]) is essentially the coding of an interval exchange transformation with exactly one connection of length 0 ending in the midpoint of the interval. In the following result we generalize a result of [5] and show that the natural coding of an interval exchange is acyclic.…”

Specular Sets

“…We say that L(T ) is the natural coding of T . which extends to an automorphism of the free group on {a, b, c} (see [10]).…”

“…Then α • V = T • α and thus L(V ) = L(T ). The interval exchange V is actually the orientation covering of the linear involution T (see [10]).…”

“…There are two interesting features of the subject of this paper. In the first place, some of the statements concerning the natural codings of linear involutions can be proved using geometric methods, as shown in a separate paper [10]. This provides an interesting interpretation of the groups playing a role in the natural codings (these groups are generated either by return words or by maximal bifix codes) as fundamental groups of some surfaces.…”