Generic coarse geometry of leaves

Abstract

A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same constants by some given set of leaves. For instance, the following properties are proved. Either all dense leaves without holonomy are equi-coarsely quasi-isometric to each other, or else there exist residually many dense leaves without holonomy such that each of them is coarsely quasi-isometric to meagerly many leaves. Assuming that the foliated space is minimal, the first of the above alternatives holds if and if the leaves without holonomy satisfy a condition called coarse quasi-symmetry. A similar dichotomy holds for the growth type of the leaves, as well as an analogous characterization of the first alternative in the minimal case, involving a property called growth symmetry. Moreover some classes of growth are shared, either by residually many leaves, or by meagerly many leaves. If some leaf without holonomy is amenable, then all dense leaves without holonomy are equi-amenable, and, in the minimal case, they satisfy a property called amenable symmetry. Residually many leaves have the same asymptotic dimension. If the foliated space is minimal, then any pair of nonempty open sets in the Higson coronas of the leaves with holonomy contain homeomorphic nonempty open subsets. Another part studies limit sets of leaves at points in their Higson corona, defined like the usual limit sets at their ends.