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

TL;DR

This paper investigates semisimple actions of automorphism groups of free groups on CAT(0) spaces, revealing fixed point properties, geometric structures like rhombic dodecahedra, and constraints on mappings to other groups.

Contribution

It characterizes fixed point behaviors of Aut(F_n) actions on CAT(0) spaces and describes new geometric actions involving rhombic dodecahedra, advancing understanding of group actions.

Findings

For n≥4, Nielsen generators have fixed points.

For n=3, actions are either fixed points or involve rhombic dodecahedron structures.

Aut(F_n) and Out(F_n) are not fundamental groups of compact Kähler manifolds.

Abstract

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 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 $\Z^4\subset Aut(F_3)$ leaves invariant an isometrically embedded copy of Euclidean 3-space on which it acts as a discrete group of translations with the rhombic dodecahedron as a fundamental 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 $n\ge 2$ then neither Aut$(F_n)$ nor Out$(F_n)$ is the fundamental group of a compact K\"ahler manifold.