Some Classes of Metric Spaces

In this chapter we describe a variety of metric spaces whose quasiisometry types include those of the leaves of compact foliated spaces. In this chapter, we recall some classes of metric spaces: graphs, metric spaces of coarse bounded geometry, and coarsely quasi-convex metric spaces. They contain the coarse quasi-isometric types of leaves of compact foliated spaces. We also introduce the class of coarsely quasi-symmetric metric spaces, which plays a relevant role in one of our main results, Theorem 1.3.