Dehn function and asymptotic cones of Abels' group

TL;DR

This paper proves that Abels' group over any nondiscrete locally compact field has a quadratic Dehn function and explores the implications for asymptotic cones and fundamental groups in various classes of groups.

Contribution

It establishes the quadratic Dehn function for Abels' group over general fields and constructs examples of groups with unusual asymptotic cone properties.

Findings

Abels' group has quadratic Dehn function over any nondiscrete locally compact field.

Existence of connected Lie groups and polycyclic groups with asymptotic cones having uncountable abelian fundamental groups.

Construction of uncountably many non-quasi-isometric solvable groups from finite characteristic cases.

Abstract

We prove that Abels' group over an arbitrary nondiscrete locally compact field has a quadratic Dehn function. As applications, we exhibit connected Lie groups and polycyclic groups whose asymptotic cones have uncountable abelian fundamental group. We also obtain, from the case of finite characteristic, uncountably many non-quasi-isometric finitely generated solvable groups, as well as peculiar examples of fundamental groups of asymptotic cones.