Date of Award
Doctor of Philosophy
Nikolay Brodskiy, Stefan Richter, Mark Hector
We begin by recalling the notion of a coarse space as defined by John Roe. We show that metrizability of coarse spaces is a coarse invariant. The concepts of bounded geometry, asymptotic dimension, and Guoliang Yu's Property A are investigated in the setting of coarse spaces. In particular, we show that bounded geometry is a coarse invariant, and we give a proof that finite asymptotic dimension implies Property A in this general setting. The notion of a metric approximation is introduced, and a characterization theorem is proved regarding bounded geometry.
Chapter 7 presents a discussion of coarse structures on the minimal uncountable ordinal. We show that it is a nonmetrizable coarse space not of bounded geometry. Moreover, we show that this space has asymptotic dimension 0; hence, it has Property A.
Finally, Chapter 8 regards coarse structures on products of coarse spaces. All of the previous concepts above are considered with regard to 3 different coarse structures analogous to the 3 different topologies on products in topology. In particular, we see that an arbitrary product of spaces with any of the 3 coarse structures with asymptotic dimension 0 has asymptotic dimension 0.
Bunn, Jared R, "Bounded Geometry and Property A for Nonmetrizable Coarse Spaces. " PhD diss., University of Tennessee, 2011.