[Paper Review] Relative free splitting and free factor complexes I: Hyperbolicity

Michael Handel, Lee Mosher|arXiv (Cornell University)|Jul 13, 2014

Geometric and Algebraic Topology23 references18 citations

TL;DR

This paper establishes the hyperbolicity of relative free splitting and free factor complexes for a group relative to a free factor system, generalizing earlier results on absolute complexes. Using a projection map from the free splitting complex to the free factor complex and applying the Kapovich–Rafi theorem on quasi-isometric embeddings, the authors prove that geodesics in the relative free splitting complex project uniformly close to geodesics in the relative free factor complex, confirming hyperbolicity in the general setting of groups with free factor systems.

ABSTRACT

We study the large scale geometry of the relative free splitting complex and the relative free factor complex of the rank $n$ free group $F_n$, relative to the choice of a free factor system of $F_n$, proving that these complexes are hyperbolic. Furthermore we present the proof in a general context, obtaining hyperbolicity of the relative free splitting complex and of the relative free factor complex of a general group $Γ$, relative to the choice of a free factor system of $Γ$. The proof yields information about coarsely transitive families of quasigeodesics in each of these complexes, expressed in terms of fold paths of free splittings.

Motivation & Objective

• To generalize the hyperbolicity results of absolute free splitting and free factor complexes to the relative setting, where a group is equipped with a free factor system.
• To define and study the relative free splitting complex $\mathcal{FS}(\Gamma;\mathcal{A})$ and relative free factor complex $\mathcal{FF}(\Gamma;\mathcal{A})$ for a group $\Gamma$ and a free factor system $\mathcal{A}$.
• To establish that these relative complexes are nonempty, connected, and hyperbolic, even when $\mathcal{A}$ is non-exceptional.
• To extend the framework to arbitrary groups, not just free groups, using the relative outer space formalism of Guirardel and Levitt.
• To prove that the projection from the relative free splitting complex to the relative free factor complex preserves quasi-isometric properties, ensuring uniform control over geodesic images.

Proposed method

• Define the relative free splitting complex $\mathcal{FS}(\Gamma;\mathcal{A})$ as the simplicial complex of equivalence classes of free splittings $T$ such that $\mathcal{F}(T) \succ \mathcal{A}$, with the collapse relation $S \succ T$ as the partial order.
• Define the relative free factor complex $\mathcal{FF}(\Gamma;\mathcal{A})$ as the geometric realization of the partial order $\sqsubset$ on free factor systems $\mathcal{B}$ satisfying $\mathcal{A} \sqsubset \mathcal{B} \neq \{[\Gamma]\}$.
• Construct a special projection map $\pi: \mathcal{FS}(\Gamma;\mathcal{A}) \to \mathcal{FF}(\Gamma;\mathcal{A})$ that sends each free splitting $T$ to its free factor system $\mathcal{F}(T)$, provided $\mathcal{F}(T) \neq \mathcal{A}$.
• Apply the Kapovich–Rafi theorem on quasi-isometric embeddings, verifying that the projection $\pi$ is Lipschitz and that images of geodesics in $\mathcal{FS}(\Gamma;\mathcal{A})$ are uniformly Hausdorff close to geodesics in $\mathcal{FF}(\Gamma;\mathcal{A})$.
• Use fold sequences and quasigeodesic reparameterization to show that any path in $\mathcal{FS}(\Gamma;\mathcal{A})$ with bounded $\mathcal{F}(T)$-diameter projects to a uniformly bounded set in $\mathcal{FF}(\Gamma;\mathcal{A})$.
• Leverage the hyperbolicity of $\mathcal{FS}(\Gamma;\mathcal{A})$ and the Lipschitz property of $\pi$ to conclude that $\mathcal{FF}(\Gamma;\mathcal{A})$ is hyperbolic via the quasi-isometric embedding theorem.

Experimental results

Research questions

• RQ1 Is the relative free splitting complex $\mathcal{FS}(\Gamma;\mathcal{A})$ hyperbolic for any proper free factor system $\mathcal{A}$ of a group $\Gamma$?
• RQ2 Does the projection from the relative free splitting complex to the relative free factor complex preserve the quasi-isometric structure of geodesics?
• RQ3 Can the hyperbolicity of the relative free factor complex $\mathcal{FF}(\Gamma;\mathcal{A})$ be established independently of the absolute case, particularly when $\mathcal{A}$ is non-exceptional?
• RQ4 How does the structure of fold sequences in free splittings relate to the diameter of their projections in the free factor complex?
• RQ5 To what extent can the results on free groups be generalized to arbitrary groups equipped with a free factor system?

Key findings

• The relative free splitting complex $\mathcal{FS}(\Gamma;\mathcal{A})$ is nonempty, connected, and hyperbolic for any proper free factor system $\mathcal{A}$ of a group $\Gamma$.
• The relative free factor complex $\mathcal{FF}(\Gamma;\mathcal{A})$ is positive dimensional, connected, and hyperbolic for any non-exceptional free factor system $\mathcal{A}$.
• The projection map $\pi: \mathcal{FS}(\Gamma;\mathcal{A}) \to \mathcal{FF}(\Gamma;\mathcal{A})$ is Lipschitz and maps any geodesic in $\mathcal{FS}(\Gamma;\mathcal{A})$ uniformly Hausdorff close to a geodesic in $\mathcal{FF}(\Gamma;\mathcal{A})$.
• The image of a geodesic under $\pi$ has uniformly bounded diameter in $\mathcal{FF}(\Gamma;\mathcal{A})$, specifically at most 1, when the endpoints are not equal to $\mathcal{A}$.
• The hyperbolicity of $\mathcal{FF}(\Gamma;\mathcal{A})$ follows from the hyperbolicity of $\mathcal{FS}(\Gamma;\mathcal{A})$ and the quasi-isometric embedding properties of $\pi$, via the Kapovich–Rafi theorem.
• The results extend beyond free groups to arbitrary groups $\Gamma$ with a free factor system $\mathcal{A}$, generalizing the framework of Guirardel and Levitt’s relative outer space.