The study of translation equivalence on integer lattices.

摘要

This paper is a contribution to the study of countable Borel equivalence relations on standard Borel spaces. We concentrate here on the study of the nature of translation equivalence on $2Zn$. We study these known hyperfinite spaces in order to gain insight into the approach necessary to classify $2Zw$ as either being hyperfinite or not. In Chapter 1, we will give the basic definitions and examples of spaces used in this work. The general construction of marker sets is developed in this work. These marker sets are used to develop several invariant tilings of the equivalence classes of $2Zn$. Some properties that are equivalent to hyperfiniteness in the space $2Zw$ are also developed. Lastly, we will give the new result that there is a continuous injective embedding from $2Z$ into $2w$.