$Z^n$-free groups are CAT(0)

TL;DR

This paper proves that groups equipped with a free -length function are CAT(0), establishing a significant link between algebraic properties and geometric group theory.

Contribution

It introduces a new class of groups with free -length functions and proves they are CAT(0), expanding understanding of geometric structures in group theory.

Findings

Groups with free -length functions are CAT(0)

Establishes a connection between algebraic length functions and geometric properties

Provides a new framework for analyzing group actions on CAT(0) spaces

Abstract

We show that every group with free $\mathbb{Z}^n$-length function is CAT(0).