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

Martin R. Bridson. "The rhombic dodecahedron and semisimple actions of Aut(Fₙ) on CAT(0) spaces." Fundamenta Mathematicae 214.1 (2011): 13-25. <http://eudml.org/doc/286121>.

@article{MartinR2011,
abstract = { 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. },
author = {Martin R. Bridson},
journal = {Fundamenta Mathematicae},
keywords = {automorphism groups of free groups; complete CAT(0) spaces; rhombic dodecahedron; actions by semisimple isometries; Nielsen generators; mapping class groups},
language = {eng},
number = {1},
pages = {13-25},
title = {The rhombic dodecahedron and semisimple actions of Aut(Fₙ) on CAT(0) spaces},
url = {http://eudml.org/doc/286121},
volume = {214},
year = {2011},
}

TY - JOUR
AU - Martin R. Bridson
TI - The rhombic dodecahedron and semisimple actions of Aut(Fₙ) on CAT(0) spaces
JO - Fundamenta Mathematicae
PY - 2011
VL - 214
IS - 1
SP - 13
EP - 25
AB - 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.
LA - eng
KW - automorphism groups of free groups; complete CAT(0) spaces; rhombic dodecahedron; actions by semisimple isometries; Nielsen generators; mapping class groups
UR - http://eudml.org/doc/286121
ER -