### University of Arkansas Topology Seminar: 3/9/2023

### Speaker: Matthew Clay

### Title: Bounded projections in the *Z*−factor graph I

Let *G* be a free product *G* = *A*_{1} * *A*_{2}* ... * *A*_{k} * *F*_{N} and let *O* be the associated outer space. 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. In this talk, we will give the context for the theorem and our intended application to the geometry of free group extensions. This is joint work with Caglar Uyanik.

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