# On the horofunction boundary of discrete Heisenberg group

## TL;DR

This paper investigates the action of the discrete Heisenberg group on its horofunction boundary, providing evidence that nilpotent groups act trivially, using a discrete isoperimetric inequality as a key tool.

## Contribution

It proves that the discrete Heisenberg group acts trivially on its horofunction boundary, supporting the conjecture for nilpotent groups.

## Key findings

- Heisenberg group acts trivially on its horofunction boundary
- Discrete isoperimetric inequality is used as a main tool
- Supports conjecture for nilpotent groups' boundary actions

## Abstract

We consider finitely generated group endowed with a word metric. The group acts on itself by isometries, which induces an action on its horofunction boundary. The conjecture is that nilpotent groups act trivially on their reduced boundary. We will show this for the Heisenberg group. The main tool will be a discrete version of the isoperimetric inequality.