The rhombic dodecahedron and semisimple actions of ${\rm{Aut}}(F_n)$ on {\rm CAT}$(0)$ spaces

Martin R. Bridson¹

¹ Mathematical Institute 24–29 St Giles' Oxford OX1 3LB, U.K.

Fundamenta Mathematicae, Tome 214 (2011) no. 1, pp. 13-25.

Résumé

We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT$(0)$ spaces. If $n\ge 4$ then each of the Nielsen generators of ${\rm Aut}(F_n)$ 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 $\mathbb Z^4\subset\mathop{\rm Aut} (F_3)$ leaves invariant an isometrically embedded copy of Euclidean 3-space $\mathbb E^3\hookrightarrow X$ 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 ${\rm Aut}(F_n)$ to mapping class groups and linear groups are obtained. If $n\ge 2$ then neither ${\rm Aut}(F_n)$ nor ${\rm{Out}}(F_n)$ is the fundamental group of a compact Kähler manifold.

Détail

DOI : 10.4064/fm214-1-2

Keywords: consider actions automorphism groups groups semisimple isometries complete cat spaces each nielsen generators aut has fixed point either each nielsen generators has nbsp fixed point else hyperbolic each nielsen generated mathbb subset mathop aut leaves invariant isometrically embedded copy euclidean space mathbb hookrightarrow which acts discrete group translations rhombic dodecahedron dirichlet domain abundance actions second kind described constraints maps aut mapping class groups linear groups obtained neither aut nor out fundamental group nbsp compact hler manifold