Topological properties of asymptotic cones

Abstract

Gromov asked whether an asymptotic cone of a finitely generated group was always simply connected or had uncountable fundamental group. We prove that Gromov's dichotomy holds for asymptotic cones with cut points as well as HNN extensions and amalgamated products where the associated subgroups are nicely embedded. We also show a slightly weaker dichotomy for multiple HNN extensions of free groups.

We define an analogue to Gromov's loop division property which we use to give a sufficient condition for an asymptotic cone of a complete geodesic metric space to have uncountable fundamental group. This is used to understand the asymptotic cones of many groups currently in the literature.

As a corollary, we show that an infinite group is virtually cyclic if and only if an asymptotic cone of the group has exactly two-ends. As well we show that in every asymptotic cone of a finitely generated group which contains a cut-point, the maximal transversal trees are universal R-trees with continuum branching at every point.