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Fixed points of finite groups of free group automorphisms. (English) Zbl 0798.20021
Dealing with the outer automorphism group \(\text{Out}(F_ n)\) of the \(n\)-generator free group \(F_ n\), M. Culler and K. Vogtmann [Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)] introduced an outer space \(X_ n\) of actions of \(F_ n\) on some metric trees upon which \(\text{Out}(F_ n)\) acts properly discontinuously. And they showed that \(X_ n\) is a finite-dimensional contractible space. R. Skora [preprint] investigated some natural compactification \(\overline{X}_ n\) of \(X_ n\) with the boundary consisting of certain actions of \(F_ n\) on \(\mathbb{R}\)-trees and proved \(\overline{X}_ n\) to be contractible. A \(\mathbb{R}\)-tree \(T\) is a metric space – with its metric denoted by \(d_ T\) – such that any two points are joined by a unique embedded arc isometric to a finite interval \(\subseteq \mathbb{R}\). Let \(G \times T \overset\circ\rightarrow T\) be a (left) action of an f.g. group \(G\) on an \(\mathbb{R}\)-tree \(T\). Such an action \((G,T)\) is called semisimple if one of the following three conditions is satisfied: (1) no finite set of ends of \(T\) is invariant under \(G\), (2) \(G\) interchanges a pair of ends of \(T\), and (3) \((G,T)\) is trivial.
The author defines for any map \(\lambda: G \to \mathbb{R}\) the weighted length function \(f_ \lambda : T \to \mathbb{R}\) to be \(\sup_{g\in G}\lambda(g) \cdot d_ T(x,g\circ x)\), the length \(l(\lambda)\) of \(\lambda\) by \(\ell(\lambda) = \inf_{x \in T} f_ \lambda(x)\) and then the characteristic set \(T_ \lambda\) of \(\lambda\) by \(T_ \lambda = \{x \in T\mid f_ \lambda(x) = \ell(\lambda)\}\). Developing F. Paulin’s ideas [Topology Appl. 32, No. 3, 197-221 (1989; Zbl 0675.20033)] based on ‘\(P\)-equivariant \(\varepsilon\)-approximations’ between compact metric spaces, the author introduces a natural topology on the space \(\mathcal H\) of actions of \(G\) on \(\mathbb{R}\)-trees and proves the following result (Th. 4.4): Let \(\mathcal H\) be the space of nontrivial semisimple actions of \(F_ n\) on \(\mathbb{R}\)-trees. Then there exists a continuous deformation \(F: X_ n \times {\mathcal H} \times I \to {\mathcal H}\) such that (1) \(F(T_ 0,T_ 1,0) = T_ 0\); (2) \(F(T_ 0,T_ 1,1) = T_ 1\); (3) \(F(T_ 0,T_ 0,t) = T_ 0\) for all \(t \in I\) and (4) \(F\) is equivariant under the diagonal action of \(\text{Out}(F_ n)\) on \(X_ n \times {\mathcal H}\). The result remains true if \(\mathcal H\) is replaced by the compactification \(\overline{X}_ n \subset {\mathcal H}\) for \(X_ n\).
An important corollary is that the subset \(X_ G\) (resp. \(\overline{X}_ G\)) of \(X_ n\) (resp. \(\overline{X}_ n\)) fixed by a finite subgroup \(G \leq \text{Out}(F_ n)\) is contractible.

MSC:
20E08 Groups acting on trees
20F65 Geometric group theory
57M05 Fundamental group, presentations, free differential calculus
20F05 Generators, relations, and presentations of groups
57S30 Discontinuous groups of transformations
20F28 Automorphism groups of groups
20E05 Free nonabelian groups
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[1] Marc Culler, Finite groups of outer automorphisms of a free group, Contributions to group theory, Contemp. Math., vol. 33, Amer. Math. Soc., Providence, RI, 1984, pp. 197 – 207. · Zbl 0552.20024 · doi:10.1090/conm/033/767107 · doi.org
[2] Marc Culler and John W. Morgan, Group actions on \?-trees, Proc. London Math. Soc. (3) 55 (1987), no. 3, 571 – 604. · Zbl 0658.20021 · doi:10.1112/plms/s3-55.3.571 · doi.org
[3] Marc Culler and Karen Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986), no. 1, 91 – 119. · Zbl 0589.20022 · doi:10.1007/BF01388734 · doi.org
[4] Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53 – 73. · Zbl 0474.20018
[5] Sava Krstić and Karen Vogtmann, Equivariant outer space and automorphisms of free-by-finite groups, Comment. Math. Helv. 68 (1993), no. 2, 216 – 262. · Zbl 0805.20030 · doi:10.1007/BF02565817 · doi.org
[6] Frédéric Paulin, The Gromov topology on \?-trees, Topology Appl. 32 (1989), no. 3, 197 – 221. · Zbl 0675.20033 · doi:10.1016/0166-8641(89)90029-1 · doi.org
[7] R. Skora, Deformations of length functions in groups, preprint. · Zbl 0607.57008
[8] M. Steiner, Ph.D. thesis, Columbia University, 1988.
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