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

Abstract

Suppose $G$ 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 $\mathcal{O}$ 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.

37 pages; v2: final version, incorporate changes suggested by referee, to appear in the Journal of Topology