On the geometry of the free factor graph for ${\rm{Aut}}(F_N)$

TL;DR

The paper analyzes the large-scale geometry of the free factor graph $\mathcal{AF}_N$ for ${\rm Aut}(F_N)$, contrasting it with the hyperbolic Out-graph. By combining a pseudo-Anosov mapping class on a surface with one boundary component and inner automorphisms given by the boundary, it constructs a natural ${\mathbb Z}^2$-subgroup whose action on $\mathcal{AF}_N$ yields quasi-isometrically embedded orbits, thereby proving $\mathcal{AF}_N$ is not Gromov-hyperbolic for $N\ge 2$. Central to the method are $b$-reduced decompositions and the invariant $[\cdot]_b$, which enable Lipschitz retractions onto ${\rm ad}_b$-orbits and robust control under change of basis. The results establish a framework for constructing quasi-flats in $\mathcal{AF}_N$ via boundary Dehn twists and pseudo-Anosov dynamics, and they raise questions about higher-rank flats and the (non)hyperbolic structure of the Aut version of free factor geometry.

Abstract

Let $Φ$ be a pseudo-Anosov diffeomorphism of a compact (possibly non-orientable) surface $Σ$ with one boundary component. We show that if $b \in π_1(Σ)$ is the boundary word, $φ\in {\rm{Aut}}(π_1(Σ))$ is a representative of $Φ$ fixing $b$, and ${\rm{ad}}_b$ denotes conjugation by $b$, then the orbits of $\langle φ, {\rm{ad}}_b \rangle\cong\mathbb{Z}^2$ in the graph of free factors of $π_1(Σ)$ are quasi-isometrically embedded. It follows that for $N \geq 2$ the free factor graph for ${\rm{Aut}}(F_N)$ is not hyperbolic, in contrast to the ${\rm{Out}}(F_N)$ case.

Figures (1)

• Figure 1: The proof of the cancellation lemma.

Theorems & Definitions (27)

• Theorem 1: Bestvina--Feighn BF
• Theorem 2
• proof
• Theorem 3
• Corollary 4
• Theorem 5
• Lemma 2.1: Whitehead's Cut-Vertex Lemma Whitehead
• Proposition 2.2
• Definition 3.1: $b$-reduced decomposition
• Remark 3.2: Geometric Interpretation