Dual automorphisms of free groups

TL;DR

This paper introduces dual automorphisms of free groups, providing an algorithm to determine how automorphisms act on cylinders in the boundary of free groups, with bounds related to Nielsen automorphisms.

Contribution

It defines dual automorphisms based on cylinders, proves bounds on their complexity, and offers an efficient algorithm for their computation.

Findings

At most 2N distinct sets U_i describe automorphism action on cylinders.

Each U_i's size is bounded by 2^t, with t related to Nielsen automorphisms.

The dual automorphism depends only on the last letter of the word.

Abstract

For any choice of a basis $\cal A$ the free group $F_N$ of finite rank $N \geq 2$ can be canonically identified with the set $F(\cal A)$ of reduced words in $\cal A\cup \cal A^{-1}$. However, such a word $w \in F(\cal A)$ admits a second interpretation, namely as cylinder $C^1_w \subset \partial F_N$. The subset of $\partial F_N$ defined by $C^1_w$ depends not only on the element of $F_N$ given by the word $w$, but also on the chosen basis $\cal A$. In particular one has in general, for $\Phi \in \Aut(F_N)$: $$\Phi(C^1_w) \neq C^1_{\Phi(w)}$$ Indeed, the image of a cylinder under an automorphism $\Phi \in \Aut(F_N)$ is in general not a cylinder, but a finite union of cylinders: $$\Phi (C^1_w)=C^{1}_U := \bigcup_{u_i \in U} C^1_{u_i}$$ In his thesis the first author has given an efficient algorithm and a formula how to determine such a (uniquely determined) finite {\em reduced} set $U = U(w) \subset F_N$. We use those to define the dual automorphism $\Phi_{\cal A}^*$ by setting $\Phi_{\cal A}^*(w) = U(w)$. \smallskip \noindent {\bf Theorem:} {\it For any $\Phi \in \Aut(F_N)$ there are at most 2N distinct finite subsets $U_i \subset F_N$ such that for any $w = y_1 ... y_r \in F_A$ there is one of them, say $U_{i(w)}$, with $$\Phi_{\cal A}^*(w) = \Phi(w) U_{i(w)}\, ,$$ and $U_{i(w)}$ depends only on the last letter $y_r \in \CA \cup \CA^{-1}$. Furthermore, the seize of each $U_{i}$ is bounded by $2^t$, where $t \geq 0$ is the number of Nielsen automorphisms in any decomposition of $\Phi$ as product of basis permutations, basis inversions and elementary Nielsen automorphisms.}