Bounded projections to the $\mathcal{Z}$-factor graph

TL;DR

This work proves a bounded-projection phenomenon for relative outer spaces and $\mathcal{Z}$-factor graphs in the setting of non-sporadic torsion-free free products $G= A_1*\cdots*A_k*F_N$. The authors develop a robust framework using Bowditch boundaries, decomposition spaces, and generalized Whitehead graphs to analyze when a non-peripheral element $g$ remains short along Grushko trees and how this constrains the projection to $\mathcal{Z}$-splitting spaces. They treat three major element-types—simple, quadratic, and $\mathcal{Z}$-simple—proving uniform diameter bounds for each case by constructing nearby splittings where $g$ is elliptic and by controlling related boundary cut-sets and edge-stabilizer lengths. The results hinge on a detailed interplay between relative hyperbolicity, boundary dynamics, and a Whitehead-graph–driven analysis of decomposition spaces, with the main bound providing a stepping stone toward a criterion for hyperbolicity of free-group extensions in forthcoming work. Overall, the paper extends systole-type projection techniques from surfaces to free products, yielding quantitative control over short conjugacy classes and their splittings in the Z-factor graph.]

Abstract

Suppose $G$ is a free product $G = A_1 * A_2* \cdots * A_k * F_N$, where each of the groups $A_i$ is torsion-free and $F_N$ is a free group of rank $N$. Let $\mathcal{O}$ be the deformation space associated to this free product decomposition. We show that the diameter of the projection of the subset of $\mathcal{O}$ where a given element has bounded length to the $\mathcal{Z}$-factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of $G$ as a hyperbolic group relative to the collection of subgroups $A_i$ together with a given non-peripheral cyclic subgroup. The main theorem is new even in the case that $G = F_N$, in which case $\mathcal{O}$ is the Culler-Vogtmann outer space. In a future paper, we will apply this theorem to study the geometry of free group extensions.

Figures (11)

• Figure 1: The quotient graph of groups $T/G$ in Example \ref{['ex:whitehead graph']}.
• Figure 2: Some translates of $T_g$ in $T$ that meet $v$ and the Whitehead graph $\mathop{\mathrm{Wh}}\nolimits_T(\mathcal{L}_g,v)$ from Example \ref{['ex:whitehead graph']}.
• Figure 3: The set-up in the proof of Lemma \ref{['lem:decomposition connected subset']}. Vertices of $T$ are filled in black, points that below in $P$ are filled in red and other midpoints of edges are filled in white. The point $q'$ may or not not belong to $P$.
• Figure 4: The graph $T_v(\mathcal{L}_g)$ in Example \ref{['ex:whitehead graph subtree']}.
• Figure 5: The set-up for the subtree $X$, shown in yellow, and some translates of $T_g$ in Example \ref{['ex:whitehead graph subtree']}.

Theorems & Definitions (78)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2: ar:Horbez17
• Definition 2.3
• Definition 3.1
• Lemma 3.2
• Example 4.1
• Definition 4.2
• Lemma 4.3