# Translation-like actions of nilpotent groups

## TL;DR

This paper establishes a new obstruction to translation-like actions between finitely generated torsion free nilpotent groups, based on differences in their Carnot completions despite having the same polynomial growth degree.

## Contribution

It introduces a novel criterion involving Carnot completions that prevents certain translation-like actions between nilpotent groups.

## Key findings

- No injective Lipschitz functions exist between groups with different Carnot completions.
- Groups with the same polynomial growth but different asymptotic cones cannot act translation-like on each other.
- Provides a new geometric obstruction in the study of nilpotent group actions.

## Abstract

We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other.