Specular sets

Abstract

We introduce specular sets. These are subsets of groups which form a natural generalization of free groups. These sets of words are an abstract generalization of the natural codings of interval exchanges and of linear involutions. We consider two important families of sets contained in specular sets: sets of return words and maximal bifix codes. For both families we prove several cardinality results as well as results concerning the subgroup generated by these sets.