Nonstandard Analysis of Graphs

نویسنده

  • F. Javier Thayer
چکیده

Intuitively, it is tempting to view a cube in R as a discrete lattice with infinitesimal edges. A similar infinitesimal picture suggests itself with other fractal spaces such as the Sierpinski gasket or the Sierpinski sponge (see [13], [5].) The purpose of this paper is to show certain classes of metric spaces characterized by volume growth properties of balls ([4],[2]) can viewed as graphs with infinitesimal edges. Our approach is based on nonstandard analysis. The main result of the paper is Theorem 10.5 which exhibits an Ahlfors regular length space as a hyperfinite graph in which the edges have infinitesimal length. The structure of the paper is as follows: the next two sections consist of reference material on nonstandard analysis and a summary of results in nonstandard measure theory. In §5, we introduce the main idea of the paper, namely the relation between polynomial growth in hyperfinite graphs and Ahlfors regularity of a naturally associated metric space. The remainder of the paper shows that the converse (suitably stated) also holds for the class of length spaces introduced by Gromov in [6]. We finish the paper with a result on lifting measure preserving automorphisms. We note that the relation between Gromov’s ideas and nonstandard analysis has been noticed before ([16]). However, the relation between polynomial growth of graphs and volume growth seems to be new.

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تاریخ انتشار 2001