Return words of linear involutions and fundamental groups

TL;DR

This paper explores the structure of return words in linear involutions, showing they form symmetric bases of free groups and their subgroups, linking geometric and algebraic properties.

Contribution

It introduces a new definition of return words for linear involutions and proves their basis properties in free groups and subgroups.

Findings

First return words form symmetric bases of free groups.

First return words form symmetric bases of finite index subgroups.

Connects geometric representations with algebraic group structures.

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 $G$.