White, Tad Fixed points of finite groups of free group automorphisms. (English) Zbl 0798.20021 Proc. Am. Math. Soc. 118, No. 3, 681-688 (1993). 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. Reviewer: U.Kaljulaid (Tartu) Cited in 1 ReviewCited in 8 Documents 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 Keywords:free group of finite rank; semisimple actions on \(\mathbb{R}\)-trees; outer automorphism group; free group; outer space; contractible space; length function; compact metric spaces; deformation; diagonal action Citations:Zbl 0589.20022; Zbl 0675.20033 PDFBibTeX XMLCite \textit{T. White}, Proc. Am. Math. Soc. 118, No. 3, 681--688 (1993; Zbl 0798.20021) Full Text: DOI References: [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 [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 [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 [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 [6] Frédéric Paulin, The Gromov topology on \?-trees, Topology Appl. 32 (1989), no. 3, 197 – 221. · Zbl 0675.20033 [7] R. Skora, Deformations of length functions in groups, preprint. · Zbl 0607.57008 [8] M. Steiner, Ph.D. thesis, Columbia University, 1988. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.