Asymptotic cones of HNN extensions and amalgamated products

TL;DR

This paper investigates the topological properties of asymptotic cones of certain groups, proving a dichotomy related to their fundamental groups, especially for HNN extensions and amalgamated products with specific subgroup embeddings.

Contribution

It establishes Gromov's dichotomy for asymptotic cones with cut points and for HNN extensions and amalgamated products under certain conditions, extending understanding of their large-scale geometry.

Findings

Gromov's dichotomy holds for asymptotic cones with cut points.

The dichotomy applies to HNN extensions and amalgamated products with well-embedded subgroups.

A weaker form of the dichotomy is shown for multiple HNN extensions of free groups.

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 asympotic 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.