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Return words of linear involutions and fundamental groups. (English) Zbl 1375.37033
The paper is devoted to the investigation of the dynamics of special maps called linear involutions. These involutions are injective piecewise isometries defined on a pair of intervals. There is a geometric representation of the considered involutions as Poincaré maps of measured foliations. Standard methods of symbolic dynamics lead to the set $$\mathcal{L}(T)$$ of finite words known as the natural coding of the linear involution $$T$$. Further, one can apply to the set $$\mathcal{L}(T)$$ standard notations from word combinatorics such as return words and prime words. Then it is proved that under suitable conditions these sets of words are symmetric bases of free group or subgroup of finite index of the free group.

##### MSC:
 37B10 Symbolic dynamics 68R15 Combinatorics on words 05A05 Permutations, words, matrices 08A50 Word problems (aspects of algebraic structures)
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