Generalizations of Coarse Properties in Large Scale Spaces

Many results in large scale geometry are proven for a metric space. However, there exists many large scale spaces that are not metrizable. We generalize several concepts to general large scale spaces and prove relationships between them. First we look into the concept of coarse amenability and other variations of amenability on large scale spaces. This leads into the definition of coarse sparsification and connections with coarse amenability. From there, we look into an equivalence of Sako's definition of property A on uniformly locally finite spaces and prove that finite coarse asymptotic definition implies it. As well, we define large scale exactness and prove implications with large scale property A and coarse amenability. We finally look into a stronger concept of bounded geometry on large scale spaces that is a coarse invariant and leads to a way to decompose large scale spaces.