Aspects of the large scale geometry of Out(Fn) - Lee Mosher

Joint IAS/PU Groups and Dynamics Seminar 4:30pm|Simonyi 101 Topic: Aspects of the large scale geometry of Out(Fn) Speaker: Lee Mosher Affiliation: Rutgers University - Newark May 06, 2025  The theory of hierarchically hyperbolic groups, due to Behrstock, Hagen, and Sisto, was developed by abstracting work of Masur and Minsky on mapping class groups. Study of the large scale geometry of the outer automorphism group Out(Fn) of a rank n free group Fn has advanced by applying hyperbolic hierarchy concepts, despite the fact that Out(Fn) is not hierarchically hyperbolic - its Dehn function grows too fast. Continuing in this vein, we discuss the Two Over All Theorem (joint work with Handel) which expresses an exponential flaring property of a ``foldable" homotopy equivalence between two rank n graphs, using distance in the free splitting complex of Fn as the argument of the exponential function. We mention several applications of that theorem to the geometry of the outer space and of the free splitting complex, including a new proof of the exponential growth of the Dehn function of Out(Fn) (joint work with Lyman). These works all apply in the more general context of outer automorphism groups relative to free factor systems.