Specular sets

TL;DR

This paper introduces specular sets, a generalization of free groups, and explores their properties, especially regarding subgroups generated by return words and bifix codes, linking to natural codings of linear involutions.

Contribution

It defines specular sets as a new class of subsets of groups, generalizing free groups, and analyzes their subgroup structures and coding properties.

Findings

Subgroups generated by return words in specular sets have specific structural properties.

Maximal bifix codes in specular sets exhibit particular subgroup behaviors.

Specular sets generalize natural codings of linear involutions, unifying different group coding frameworks.

Abstract

We introduce the notion of specular sets which are subsets of groups called here specular and which form a natural generalization of free groups. These sets are an abstract generalization of the natural codings of linear involutions. We prove several results concerning the subgroups generated by return words and by maximal bifix codes in these sets.