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Projection in the sphere complex

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Lucas Sabalka (SUNY Binghamton)

Abstract:
The outer automorphism group Out(F_n) of the free group shares much in common with the mapping class group of a surface: often, a result, object, or approach known for one of these groups can be translated for the other group. One of the key objects in the study of the mapping class group of a surface is the curve complex. One important use of the curve complex is in understanding the behavior of geodesics in Teichmuller space, via the hierarchy machinery of Masur and Minsky. Key ingredients in the hierarchy include the hyperbolicity of the curve complex and the projection of points in the curve complex to the curve complex of a subsurface. Recent efforts have been aimed at translating the Masur-Minsky approach to the study of Out(F_n). In the past six months, papers have been posted to the ArXiv showing the factor complex and the splitting complex (also known as the sphere complex) are both hyperbolic. In this talk, we discuss a possible definition for the other key ingredient -- projection to a `submanifold' -- for the sphere complex.