Jensen, Craig A. Contractibility of fixed point sets of auter space. (English) Zbl 0997.20042 Topology Appl. 119, No. 3, 287-304 (2002). By a construction of A. Hatcher and K. Vogtmann [J. Lond. Math. Soc., II. Ser. 58, No. 3, 633-655 (1998; Zbl 0922.57001)], the automorphism group \(\operatorname{Aut}(F_n)\) of a free group \(F_n\) of rank \(n\) acts with finite stabilizers on a contractible space \(X_n\) (a spine of “auter space”), obtained as the geometric realization of a suitable poset of equivalence classes of marked graphs with fundamental group \(F_n\). The main result of the present paper states that, for any finite subgroup \(G\) of \(\operatorname{Aut}(F_n)\), the fixed point subcomplex \(X_n^G\) of \(G\) is contractible. The proof generalizes methods of an analogous result by S. Krstic and K. Vogtmann [Comment. Math. Helv. 68, No. 2, 216-262 (1993; Zbl 0805.20030)] for the action of the outer automorphism group \(\text{Out}(F_n)\) on “outer space” (obtained also by T. White [Proc. Am. Math. Soc. 118, No. 3, 681-688 (1993; Zbl 0798.20021)]). The result is motivated by the fact that it allows one to compute the cohomology of normalizers or centralizers of finite subgroups of \(\operatorname{Aut}(F_n)\) based on their actions on fixed point subcomplexes (for example in a paper by the author [in J. Pure Appl. Algebra 158, No. 1, 41-81 (2001; Zbl 0977.20042)]). Reviewer: Bruno Zimmermann (Trieste) Cited in 5 Documents MSC: 20F65 Geometric group theory 20F28 Automorphism groups of groups 20J05 Homological methods in group theory 57M07 Topological methods in group theory Keywords:automorphism groups of free groups; outer space; auter space; contractible complexes; outer automorphism groups; cohomology Citations:Zbl 0922.57001; Zbl 0805.20030; Zbl 0798.20021; Zbl 0977.20042 PDFBibTeX XMLCite \textit{C. A. Jensen}, Topology Appl. 119, No. 3, 287--304 (2002; Zbl 0997.20042) Full Text: DOI arXiv References: [1] Alperin, R.; Bass, H., Length functions of group actions on \(Λ\)-trees, (Gersten, S. M.; etal., Combinatorial Group Theory and Topology (1987), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ), 265-378 · Zbl 0978.20500 [2] Baumslag, G.; Taylor, T., The center of groups with one defining relator, Math. Ann., 175, 315-319 (1968) · Zbl 0157.34901 [3] Culler, M., Finite groups of outer automorphisms of a free group, (Contemp. Math., 33 (1984), American Mathematical Society: American Mathematical Society Providence, RI), 197-207 [4] Culler, M.; Vogtmann, K., Moduli of graphs and automorphisms of free groups, Invent. Math., 84, 91-119 (1986) · Zbl 0589.20022 [5] Hatcher, A., Homological stability for automorphism groups of free groups, Comment. Math. Helv., 70, 39-62 (1995) · Zbl 0836.57003 [6] Hatcher, A.; Vogtmann, K., Cerf theory for graphs, J. London Math. Soc. (2), 58, 3, 633-655 (1998) · Zbl 0922.57001 [7] C.A. Jensen, Cohomology of \(Aut F_n \); C.A. Jensen, Cohomology of \(Aut F_n \) [8] Jensen, C. A., Cohomology of \(Aut (F_n )\) in the \(p\)-rank two case, J. Pure Appl. Algebra, 158, 41-81 (2001) · Zbl 0977.20042 [9] Krstic, S., Actions of finite groups on graphs and related automorphisms of free groups, J. Algebra, 124, 119-138 (1989) · Zbl 0675.20025 [10] Krstic, S.; Vogtmann, K., Equivariant outer space and automorphisms of free-by-finite groups, Comment. Math. Helv., 68, 216-262 (1993) · Zbl 0805.20030 [11] Lyndon, R.; Schupp, P., Combinatorial Group Theory (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0368.20023 [12] Quillen, D., Homotopy properties of the poset of \(p\)-subgroups of a finite group, Adv. Math., 28, 101-128 (1978) · Zbl 0388.55007 [13] Smillie, J.; Vogtmann, K., A generating function for the Euler characteristic of \(Out (F_n )\), J. Pure Appl. Algebra, 44, 329-348 (1987) · Zbl 0616.20009 [14] Smillie, J.; Vogtmann, K., Automorphisms of graphs, \(p\)-subgroups of \(Out (F_n )\) and the Euler characteristic of \(Out (F_n )\), J. Pure Appl. Algebra, 49, 187-200 (1987) · Zbl 0641.20029 [15] White, T., Fixed points of finite groups of free group automorphisms, Proc. Amer. Math. Soc., 118, 681-688 (1993) · Zbl 0798.20021 [16] Zimmerman, B., Über Homöomorphismen \(n\)-dimensionaler Henkelkörper und endliche Erweiterungen von Schottky-Gruppen, Comment. Math. Helv., 56, 424-486 (1981) [17] Kropholler, P. J.; Mislin, G., Groups acting on finite-dimensional spaces with finite stabilizers, Comment. Math. Helv., 73, 122-136 (1998) · Zbl 0927.20033 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.