On the dynamics and the fixed subgroup of a free group automorphism
The study of the automorphism group Aut \(F_ n\) of a free group \(F_ n\) by J. H. C. Whitehead used topological methods up to dimension 4. Later his results were extended and reproven in more algebraic and combinatorial terms. But recently several authors have taken up Whitehead's original ideas again and combined them with Gersten's result that rank Fix(\(\phi)\) \((\phi \in Aut(F_ n))\) is finite, in particular \textit{R. Z. Goldstein} and \textit{E. C. Turner} [Invent. Math. 78, 1-12, (1984; Zbl 0548.20016); Bull. Lond. Math. Soc. 18, 468-470 (1986; Zbl 0576.20016)]. Goldstein and Turner introduced a graph D(\(\phi)\) for \(\phi\in Aut F_ n\), whose vertices correspond to elements of \(F_ n\). Two vertices v, w are connected by an edge, if \(w=x^{-1}v\phi (x)\), where x is an element (or its inverse) of a given standard basis of \(F_ n\). Cohen and Lustig refine this method by orienting all edges of the first barycentric subdivision of D(\(\phi)\); thus the original edges become ``ordinary'', ``attractive'', or ``repulsive''. Studying the dynamics which results from these labels on D(\(\phi)\) (in particular, closed oriented paths and attractive rays towards infinity), they obtain remarkable geometric resp. algebraic formulae for the determination of rank Fix(\(\phi)\). Moreover they deduce new estimates which can be read off in practically given situations, for instance, if \(\phi\) sends each element of the standard basis to a word containing these generators only with positive exponents. As an example, Fix(\(\phi)\) turns out to be trivial for \(\phi\) : F(a,b,c)\(\to F(a,b,c)\) given by \(\phi (a)=b\), \(\phi (b)=c\), \(\phi (c)=ab\), a question raised by Stallings. The material of this paper has already been incorporated in \textit{D. E. Cohen's} new textbook: Combinatorial group theory, a topological approach (LMS, Student Texts, 14) [for a review of an earlier version see Zbl 0389.20024]. It is part of a present broader stream (where also work of Bestvina and Handel belongs to) to unravel the structure of Aut \(F_ n\) and Out \(F_ n\) by geometric methods in analogy to that of the mapping class group of surfaces.
fixed points of automorphisms of free groups
scientific article; zbMATH DE number 4111596