Homological Growth of Groups and Aspherical Manifolds

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Topology is often described as doing geometry on a rubber sheet; spaces are considered equivalent if one can be transformed into the other without cutting or gluing. Homology is a fundamental algebraic invariant which can distinguish spaces up to this equivalence. For instance, the homology of the surface of a donut is different than that of the surface of a donut hole, and this implies that turning one of these surfaces into the other requires some cutting or gluing. Homological invariants are generally difficult to compute, especially in higher dimensions. Instead of computing them exactly, this project addresses questions about the growth rate of these invariants for naturally occurring sequences of spaces. The proposed research concentrates on several long-standing conjectures on the topology of aspherical manifolds. This project will also promote graduate and undergraduate education through mentoring and outreach. More specifically, the primary goal of the research program is to study the growth of homological invariants in a residual tower of finite regular covers of an aspherical complex. The underlying motivation is a conjecture that linear growth of rational Betti numbers in such a tower should obstruct these complexes being homotopy equivalent to manifolds of a certain dimension. The project will explore this conjecture and variants of it with rational homology replaced with mod p or integral homology. For instance, the PI plans to construct Gromov hyperbolic groups where the growth rate of Betti numbers depends on the field of coefficients, and locally CAT(-1), odd-dimensional manifolds with linear growth of mod p Betti numbers. Such manifolds will not virtually fiber over a circle, and the PI will further explore the connection between vanishing homological growth and the Bieri-Neumann-Strebel invariants (which encode algebraic analogues of such fibering). The PI will also study manifold thickenings of aspherical complexes where this homological growth vanishes and other obstructions from coarse geometry arise. This project is jointly funded by Topology and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.