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

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Bridson M.

Fundamenta Mathematicae

2011

tom 214

nr 1

13-25

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.

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1 Fundamenta Mathematicae

1 Bridson M.

1 2011

Wyniki wyszukiwania

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