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An index for counting fixed points of automorphisms of free groups. (English) Zbl 0946.20010
If \(\alpha\) is an automorphism of the free group \(F_n\) of rank \(n\), then the fixed subgroup \(\text{Fix }\alpha=\{g\in F\mid\alpha(g)=g\}\) of \(\alpha\) has rank at most \(n\). This is a well known theorem proved by M. Bestvina and M. Handel [Ann. Math., II. Ser. 135, No. 1, 1-51 (1992; Zbl 0757.57004)]. In this present paper, the authors by using \(\mathbb{R}\)-trees improve this result by proving that for an automorphism \(\alpha\) of \(F_n\) then \[ \text{rk}(\text{Fix }\alpha)+\tfrac 12 a(\alpha)\leq n. \] The number \(a(\alpha)\), depending on the automorphism \(\alpha\), is defined in terms of the boundary \(\delta F_n\) of \(F_n\), i.e., the set of infinite reduced words. The action of \(\alpha\) extends to this boundary \(\delta F_n\) and the study of fixed points in \(\delta F_n\) gives rise to attracting or repelling points. Now, the set of attracting points is partitioned into equivalent classes under an equivalence defined in a simple manner and the number of these equivalence classes, proven finite by D. Cooper [J. Algebra 111, 453-456 (1987; Zbl 0628.20029)] is the number \(a(\alpha)\). If the subgroup \(\text{Fix }\alpha\) is trivial, then they get as a corollary that \(\alpha\) fixes at most \(4n\) ends of \(F_n\).
The treatment is geometric by considering an \(\alpha\)-invariant \(\mathbb{R}\)-tree \(T\) and analyzing relating properties of \(\alpha\) to geometric properties of \(T\). They also define and discuss informally two indices, one for \(\alpha\), \[ \text{ind}(\alpha)=\text{rk}(\text{Fix }\alpha)+\tfrac 12 a(\alpha)-1 \] and one for the outer automorphism \(\Phi\) appearing in their supporting results. There is a reference list of 31 items.

MSC:
20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
20E08 Groups acting on trees
57M07 Topological methods in group theory
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