%0 Book Section
%T Dual automorphisms of free groups
%+ Laboratoire d'Analyse, Topologie, Probabilités (LATP)
%+ Institut de Mathématiques de Marseille (I2M)
%A Ibrahim, Fedaa
%A Lustig, Martin
%@ 978-3-319-05487-2
%B Extended Abstracts Fall 2012 (Automorphisms of Free Groups)
%I Springer International Publishing
%S Trends in Mathematics
%V 1
%P 59-62
%8 2014
%D 2014
%Z 1306.5696
%R 10.1007/978-3-319-05488-9_11
%K automorphisms of free groups
%K cylinders
%K currents
%K growth rate
%Z 20F; 20E, 57M
%Z Mathematics [math]/Group Theory [math.GR]Book sections
%X 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.}
%G English
%L hal-01318915
%U https://hal.science/hal-01318915
%~ LATP
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-