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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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References:
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