Asymptotic cones of HNN extensions and amalgamated products

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.