We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT(0) spaces. If n ≥ 4 then each of the Nielsen generators of Aut(Fₙ) has a fixed point. If n = 3 then either each of the Nielsen generators has a fixed point, or else they are hyperbolic and each Nielsen-generated ℤ⁴ ⊂ Aut(F₃) leaves invariant an isometrically embedded copy of Euclidean 3-space 𝔼³ ↪ X on which it acts as a discrete group of translations with the rhombic dodecahedron as a Dirichlet domain. An abundance of actions of the second kind is described. Constraints on maps from Aut(Fₙ) to mapping class groups and linear groups are obtained. If n ≥ 2 then neither Aut(Fₙ) nor Out(Fₙ) is the fundamental group of a compact Kähler manifold.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-1-2,
author = {Martin R. Bridson},
title = {The rhombic dodecahedron and semisimple actions of Aut(Fn) on CAT(0) spaces},
journal = {Fundamenta Mathematicae},
volume = {215},
year = {2011},
pages = {13-25},
zbl = {1260.20063},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-1-2}
}
Martin R. Bridson. The rhombic dodecahedron and semisimple actions of Aut(Fₙ) on CAT(0) spaces. Fundamenta Mathematicae, Tome 215 (2011) pp. 13-25. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm214-1-2/