Dimension and Fundamental Groups of Asymptotic Cones

Abstract

Asymptotic cones were first used by Gromov in [6], where he constructed limit spaces of nilpotent groups in order to prove that groups with polynomial growth are virtually nilpotent. Gromov does not use the term asymptotic cone, which was introduced later by Van den Dries and Wilkie in [12], when they gave a nonstandard interpretation of Gromov's results. Ultrafilters appear in [12] for the first time in this context. Later, Gromov gave an extensive treatment of asymptotic cones in [7]. Since then several authors have used asymptotic cones to obtain interesting results, for instance, in identifying quasi-isometry classes of 3-manifolds [8] or relating asymptotic cones with Dehn functions of finitely presented groups (see [2] and [11], the results of which are stated in Section 2). The purpose of this paper is to develop some of the results stated in [7], in particular those describing the asymptotic cone of the Baumslag–Solitar groups and of Sol. According to [2], these spaces are not simply connected, since their Dehn functions are exponential, so our primary goal is to study their fundamental groups. It will be proved that these fundamental groups are uncountable and nonfree (Section 9), by constructing subgroups isomorphic to the fundamental group of the Hawaiian earring. These subgroups are constructed by finding subspaces in the asymptotic cones which are homotopically equivalent to the Hawaiian earring, and which induce injections in the fundamental group level (Section 8). Crucial to the proof of these facts is the computation of the covering dimension of these asymptotic cones (Section 7), which is done using a more general theorem on dimensions of spaces which admit certain maps into well-known spaces (Section 6).