On the geometry of free nilpotent groups

TL;DR

This paper explores the geometric structure of free nilpotent groups, providing a criterion based on Cayley graph analysis to solve the word problem for these groups.

Contribution

It introduces a geometric criterion for the word problem in finitely generated free nilpotent groups, linking algebraic equivalence to Cayley graph behavior.

Findings

Established a geometric criterion for the word problem

Connected group elements' equivalence to Cayley graph analysis

Enhanced understanding of nilpotent group geometry

Abstract

In this article, we study geometric properties of nilpotent groups. We find a geometric criterion for the word problem for the finitely generated free nilpotent groups. By geometric criterion, we mean a way to determine whether two words represent the same element in a free nilpotent group of rank $r$ and class $k$ by analyzing their behavior on the Cayley graph of the free nilpotent group of rank $r$ and class $k-1$.