Abstract
We investigate the shifts associated with natural codings of linear involutions. We deduce, from the geometric representation of linear involutions as Poincaré maps of measured foliations, a suitable definition of return words which yields that the set of return words to a given word is a symmetric basis of the free group on the underlying alphabet,$A$. The set of 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$G$.
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