## Sprache:

SFB 878
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For $n\geq 3$ we study global fixed points for isometric actions of the automorphism group of a free group of rank n on complete d-dimensional CAT(0) spaces. We prove that whenever the automorphism group of a free group of rank n acts by isometries on complete d-dimensional CAT(0) space such that $d<2\cdot\lfloor\frac{n}{3}\rfloor$ it must fix a point. This property has implications for irreducible representations of the automorphism group of a free group of rank n. For $n\geq 4$ we obtain similar results for the unique subgroup of index two in the automorphism group of a free group.