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Localization Of Large Scale Structures, Ryan James Jensen
Localization Of Large Scale Structures, Ryan James Jensen
Doctoral Dissertations
We begin by giving the definition of coarse structures by John Roe, but quickly move to the equivalent concept of large scale geometry given by Jerzy Dydak. Next we present some basic but often used concepts and results in large scale geometry. We then state and prove the equivalence of various definitions of asymptotic dimension for arbitrary large scale spaces. Some of these are generalizations of asymptotic dimension for metric spaces, and many of the proofs are new. Particularly useful in proving the equivalences of the various definitions is the notion of partitions of unity, originally set forth by Jerzy …
Bounded Geometry And Property A For Nonmetrizable Coarse Spaces, Jared R Bunn
Bounded Geometry And Property A For Nonmetrizable Coarse Spaces, Jared R Bunn
Doctoral Dissertations
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 …