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

Suppose that $G$ is a free product $G=A_1\ast A_2\ast \cdots \ast A_k\ast F_N$, where each of the groups $A_i$ is torsion-free and $F_N$ is a free group of rank $N$. Let $O$ be the deformation space associated to this free product decomposition. We show that the diameter of the projection of the subset of $O$ where a given element has bounded length to the $Z$-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 nonperipheral cyclic subgroup. The main theorem is new even in the case that $G=F_N$, in which case $O$ is the Culler–Vogtmann outer space. In a future paper, we will apply this theorem to study the geometry of free group extensions.