Classifying spaces for proper actions and almost-flat manifolds

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In this research, we will combine techniques from Geometric Group Theory, Topololgy, and Geometry to work on two objectives. In the last twenty years, non-positively curved spaces and groups have been at the forefront of Geometric Group Theory and Topology. Their importance is underlined by I. Agol's breakthrough solution of the Virtual Haken Conjecture of Thurston using the machinery of non-positively curved cube complexes developed by D. Wise. Also, in the last decade, the Baum-Connes and the Farrell-Jones Conjectures have been verified for many (non-positively curved) classes of groups, paving the way for computations in algebraic K- and L-theories via their classifying spaces. These conjectures connect many different fields of mathematics and have far reaching applications in Topology, Analysis, and Algebra. The time is therefore right to investigate (finiteness) properties of such groups and to construct models for classifying spaces for proper actions with geometric properties that are suitable for computations. Our first objective is to construct such models for classifying spaces of proper actions for some important classes of groups such as Coxeter groups and the outer automorphism group of right-angled Artin groups, and to investigate Brown's conjecture.Our second objective is on almost-flat manifolds. These manifolds are a generalisation of flat manifolds introduced by M. Gromov. They occur naturally in the study of Riemannian manifolds with negative sectional curvature and play a key role in the study of collapsing manifolds with uniformly bounded sectional curvature. The characteristic properties of these manifolds that we will investigate such as Spin structures and cobordisms play an integral part in modern manifold theory. Spin structures have many applications in Quantum Field Theory and in Mathematical Physics. In particular, the existence of a Spin structure on a smooth orientable manifold allows one to define spinor fields and a Dirac operator which can be thought of as the square root of the Laplacian. Dirac operator is essential in describing the behaviour of fermions in Particle Physics. It is also an important invariant in Pure Mathematics arising in Atiyah-Singer Index Theorem, Connes's Noncommutative Differential Geometry, the Schrodinger-Lichnerowicz formula, Kostant's cubic Dirac operator, and many other areas. The methods by which we propose to study almost-flat manifolds arise from the interactions between Geometry/Topology and Group Theory. This is largely due to the fact that the topology of these manifolds is completely classified by their fundamental groups.