Abstract
Suppose G is a free product G=A1∗A2∗⋯∗Ak∗FN, where each of the groups Ai is torsion-free and FN 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 Ai together with a given non-peripheral cyclic subgroup. The main theorem is new even in the case that G=FN, 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.
https://arxiv.org/abs/2306.17664