# The $L^2$-torsion polytope of amenable groups

## TL;DR

This paper introduces the concept of groups of polytope class and demonstrates that torsion-free amenable groups satisfying the Atiyah Conjecture have this property, leading to homotopy invariance of the $L^2$-torsion polytope.

## Contribution

It defines groups of polytope class and proves their properties for certain amenable groups, extending understanding of $L^2$-torsion invariants.

## Key findings

- Amenable groups satisfying the Atiyah Conjecture are of polytope class.
- Homotopy invariance of the $L^2$-torsion polytope is established for these groups.
- The $L^2$-torsion polytope vanishes if the group contains a non-abelian elementary amenable normal subgroup.

## Abstract

We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the $L^2$-torsion polytope among $G$-CW-complexes for these groups. As another application we prove that the $L^2$-torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.