[Paper Review] The rhombic dodecahedron and semisimple actions of Aut(F_n) on CAT(0) spaces

Martin R. Bridson|arXiv (Cornell University)|Feb 28, 2011

Geometric and Algebraic Topology4 citations

TL;DR

This paper investigates semisimple isometric actions of Aut(Fₙ) on complete CAT(0) spaces, proving that for n ≥ 4, all Nielsen generators fix a point. For n = 3, either all Nielsen generators fix a point or each generates a ℤ⁴ acting discretely on a 3-flat with the rhombic dodecahedron as a fundamental domain. The work establishes that Aut(Fₙ) and Out(Fₙ) are not fundamental groups of compact Kähler manifolds for n ≥ 2.

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.

Motivation & Objective

• To understand the geometric constraints on actions of Aut(Fₙ) on CAT(0) spaces via semisimple isometries.
• To determine whether Nielsen generators of Aut(Fₙ) act elliptically (with fixed points) or hyperbolically (with translation axes).
• To classify the possible actions of Aut(F₃) on CAT(0) spaces, particularly focusing on ℤ⁴ subgroups generated by Nielsen transformations.
• To derive topological and representation-theoretic consequences, especially regarding Kähler groups and linear representations.
• To establish that Aut(Fₙ) and Out(Fₙ) are not fundamental groups of compact Kähler manifolds for n ≥ 2.

Proposed method

• Use of the Karlsson-Margulis lemma to analyze translation lengths of isometries in CAT(0) spaces.
• Application of Gersten’s idea to show that for n ≥ 4, Nielsen generators must fix points in semisimple actions.
• Geometric analysis of ℤ⁴ subgroups generated by Nielsen transformations in Aut(F₃), showing they act discretely on isometrically embedded ℝ³ with rhombic dodecahedron fundamental domains.
• Use of the rhombic dodecahedron as a Dirichlet domain to characterize the action of Nielsen ℤ⁴ subgroups in the n = 3 case.
• Leveraging the rigidity of the standard representation Aut(Fₙ) → GL(n, ℤ) and its implications for representation theory.
• Application of Simpson’s non-deformation result and Hodge-theoretic obstructions to prove that Aut(Fₙ) and Out(Fₙ) are not Kähler groups.

Experimental results

Research questions

• RQ1 For n ≥ 4, do all Nielsen generators of Aut(Fₙ) fix a point in any semisimple isometric action on a complete CAT(0) space?
• RQ2 In the case n = 3, can Nielsen transformations act hyperbolically, and if so, what geometric structure is preserved by their ℤ⁴ subgroups?
• RQ3 Is there a uniform geometric invariant—such as a fundamental domain—associated with the action of Nielsen ℤ⁴ subgroups in Aut(F₃)?
• RQ4 What constraints do actions on CAT(0) spaces impose on homomorphisms from Aut(Fₙ) to mapping class groups or linear groups?
• RQ5 Are Aut(Fₙ) and Out(Fₙ) fundamental groups of compact Kähler manifolds for n ≥ 2?

Key findings

• For n ≥ 4, every Nielsen generator of Aut(Fₙ) fixes a point in any semisimple isometric action on a complete CAT(0) space.
• For n = 3, either all Nielsen generators fix a point, or each Nielsen ℤ⁴ subgroup acts discretely on an isometrically embedded ℝ³ with the rhombic dodecahedron as a fundamental domain.
• An abundance of hyperbolic actions of Aut(F₃) exists, arising from linear representations and the fact that Out(F₃) has a finite-index subgroup mapping onto a non-abelian free group.
• When n ≥ 6, all Nielsen generators have zero translation length in any isometric action on a complete CAT(0) space, meaning they are either elliptic or neutral parabolic.
• The standard representation of Aut(Fₙ) on H₁(Fₙ, ℤ) is rigid, and this rigidity implies that Aut(Fₙ) and Out(Fₙ) cannot be Kähler groups for n ≥ 2.
• The image of any representation Φ: Aut(Fₙ) → SL(d, ℝ) with n ≥ 6 must have eigenvalues of modulus 1, due to the zero translation length of Nielsen generators.