Title: Homology Growth in Towers and Aspherical Manifolds
Given a space and a tower of covering spaces, a natural game is to take a classical homological invariant and study its growth as one goes up the tower. If the covers are regular and residual (the intersection of the corresponding subgroups is the identity), one hopes that these limit towards an invariant of the universal cover with an analytic flavor. This turns out to be true for rational homology, and there has been a lot of recent work extending this to F_p-homology and torsion in integral homology, though the whole story remains conjectural. I'll survey various conjectures about rational/F_p homology growth and torsion growth in these covers. We'll then discuss constructions of closed aspherical manifolds that have F_p homology growth outside of the middle dimension, which disproves an F_p-version of a conjecture of Singer.
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