Relative free splitting and free factor complexes I: Hyperbolicity

TL;DR

The paper proves that the relative free splitting complex $\mathcal{FS}(\Gamma;\mathcal{A})$ and the relative free factor complex $\mathcal{FF}(\Gamma;\mathcal{A})$ are hyperbolic for any finitely generated group $\Gamma$ relative to a free factor system $\mathcal{A}$, with parallel results for $F_n$ relative to $\mathcal{A}$ and general $\Gamma$. It adapts the Masur–Minsky framework to the relative setting by constructing coarse-path families via Stallings fold paths, defining free splitting units, and establishing projection diagrams to verify the Coarse Retract, Coarse Lipschitz, and Strong Contraction Axioms; the core technical tool is the Big Diagram argument, which translates combinatorial fold behavior into global hyperbolicity constants that depend only on the corank and the size of $\mathcal{A}$. The work also proves uniform quasigeodesic parameterizations of fold paths using free splitting units and shows that the relative free factor complex inherits hyperbolicity via a Kapovich–Rafi transfer from the relative splitting complex, with the projection map controlling geodesic behavior. Together, these results provide a robust, deformation-space–inspired framework for understanding the large-scale geometry of outer automorphism groups of freely decomposable groups, and establish a foundation for analyzing dynamics of $\mathrm{Out}(\Gamma;\mathcal{A})$ in both absolute and relative contexts.

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.

Figures (10)

• Figure 1: The Extension Lemma \ref{['LemmaExtension']} shows that for each extension $\textcolor{red}{\mathcal{A}} \sqsubset \mathcal{A}'$ of free factor systems of $\Gamma$, and for any realization of $\mathcal{A}'$, there exists a free factorization with terms as depicted which simultaneously incorporates the following: the given realization of $\mathcal{A}' = \{[A'_1],\ldots,[A'_K]\}$ with cofactor $B'$; for each $j \le J$ a realization of a free factor system of $A'_j$, namely $\mathcal{A}'_j = \{[\textcolor{red}{A_{j1}}],\ldots,[\textcolor{red}{A_{jk_j}}]\}$, with cofactor $B_j$; and a realization of $\textcolor{red}{\mathcal{A}} = \{[\textcolor{red}{A_{11}}],\ldots,[\textcolor{red}{A_{1k_1}}],\ldots\ldots,[\textcolor{red}{A_{1J}}],\ldots,[\textcolor{red}{A_{1k_J}}]\}$ with cofactor $B = B_1 * \cdots * B_J * A'_{J+1} * \cdots * A'_K * B'$. In fact these conclusions can be reformulated even when $\mathcal{A}'$ is only a weak free factor system, but one then deduces that $\mathcal{A}'$ is actually a (strong) free factor system.
• Figure 2: An augmented projection diagram of depth $J$ from $T$ to $S_I\mapsto\cdots\mapsto S_K$.
• Figure 3: The Big Diagram, Step 0. Certain columns $L=L_0$, $L_1$, … are emphasized, using free splitting units along the fold path $T^0_J \to \cdots \to T^0_L$. As the Big Diagram evolves, and up until nearly the end of the evolution, the original projection diagram atop which the diagram is built, which involves the $S'$ and $S$ rows, will not change. Those rows will be suppressed in the meantime, returning only in the Penultimate Diagram of Figure \ref{['FigurePentDiagram']}.
• Figure 4: Factoring the combing rectangle between rows $2,3$ and columns $0,\ldots,L_1$.
• Figure 5: The Big Diagram, step 0.1

Theorems & Definitions (94)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4: Free factor systems.