Rectifying separated nets

Abstract

In our earlier paper [BuK] (see also McMullen’s paper [M]), we constructed examples of separated nets in the plane R2 which are not biLipschitz equivalent to the integer lattice Z2. These examples gave a negative answer to a question raised by H. Furstenberg and M. Gromov. Furstenberg asked this question in connection with Kakutani equivalence for R2-actions, [F]. Return times for a section of an R2-action form a separated net, and to represent the returns of an R2-action by a Z2-action, one has to have a biLipschitz identification of the return times for each point with Z2 (depending measurably on the point). As was pointed out to us by A. Katok, one can use a standard construction of R2-actions to represent our example as the set of return times for points from a set of positive measure, thus showing that not every section can be used (it is worth mentioning here that an old result of Katok [K] asserts that every R 2-action admits a section whose return times are biLipschitz equivalent to Z2). Gromov’s motivation for the question came from large scale geometry, and the definition of quasi-isometries. Two metric spaces are quasiisometric if they contain biLipschitz equivalent separated nets; hence one would like to know if the choice of separated net matters, and if a given space can contain nets which are not biLipschitz equivalent. This question is particularly interesting for spaces with cocompact isometry groups. The counterexample in [BuK] was based on a counterexample to another question which had been posed by J. Moser and M. Reimann in the 60’s, namely whether every positive continuous function on the plane is locally the Jacobian of a biLipschitz homeomorphism. Using well-known properties of quasi-conformal homeomorphisms, one can actually show that any