The rhombic dodecahedron and semisimple actions of Aut(F_n) on CAT(0) spaces

摘要

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(F_n) 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 Z~4 ∈ Aut(F_3) leaves invariant an isometrically embedded copy of Euclidean 3-space E~3 → 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(F_n) to mapping class groups and linear groups are obtained. If n > 2 then neither Aut(F_n) nor Out(F_n) is the fundamental group of a compact Kaler manifold.