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The Two-over-All Theorem and its Applications

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Lee Mosher (Rutgers)

Abstract: A standard application of the theory of train track maps to a fully irreducible element phi of Out(F_n) produces a train track representative of phi, which is a certain self-homotopy equivalence of a rank n graph G representing phi, such that the sequence of iterates of each edge is an exponentially growing sequence of paths in G without backtracking. I will describe the ``Two-Over-All’’ Theorem, an aperiodic generalization of this exponential growth statement, applying to any Stallings fold path in the free splitting complex of F_n having sufficiently large diameter, and with uniform exponential growth behavior as a function of diameter. I will also describe applications to the geometry of outer space, and to quantitative estimates on translation numbers of elements of Out(F_n) acting on the free splitting complex. These results all hold in the more general context of the outer automorphism group of a free group relative to a free factor system, including for example the outer automorphism group of a free product of finite groups. These works are all joint with Michael Handel. I will also discuss joint applications with Rylee Lyman regarding exponential lower bounds on Dehn functions of this same class of outer automorphism groups.