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

TL;DR

This paper proves that projections of bounded-length elements in a deformation space related to free products have uniformly bounded diameter in the $\\mathcal{Z}$-factor graph, with implications for understanding free group extensions.

Contribution

It establishes a uniform bound on the diameter of projections in the $\\mathcal{Z}$-factor graph for free products, extending known results to new settings including free groups.

Findings

Bounded projections in the $\\mathcal{Z}$-factor graph depend only on length bounds.

Analysis of the boundary of hyperbolic groups relative to subgroups is key.

Main theorem applies even to free groups, generalizing previous results.

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.