Cylinders, multi-cylinders and the induced action of $Aut(F_n)$

TL;DR

This paper studies how automorphisms of free groups transform cylinders and double cylinders, providing algorithms and formulas to describe these images as finite unions of cylinders, advancing understanding of free group automorphisms.

Contribution

It establishes that automorphisms map cylinders and double cylinders to finite unions of cylinders and provides explicit algorithms and formulas for these transformations.

Findings

Automorphisms map cylinders to finite unions of cylinders.

Algorithms are provided to compute the images of cylinders under automorphisms.

Formulas are derived for the precise description of these images.

Abstract

A cylinder $C^1_u$ is the set of infinite words with fixed prefix $u$. A double-cylinder $C^2_{[1,u]}$ is "the same" for bi-infinite words. We show that for every word $u$ and any automorphism $\varphi$ of the free group $F$ the image $\varphi(C^1_u)$ is a finite union of cylinders. The analogous statement is true for double cylinders. We give (a) an algorithm, and (b) a precise formula which allows one to determine this finite union of cylinders.