Assuad-Nagata dimension of nilpotent groups with arbitrary left invariant metrics

TL;DR

This paper demonstrates that countable groups with non-locally finite centers can be equipped with proper, left-invariant metrics resulting in infinite Assouad-Nagata dimension, addressing open problems in geometric group theory.

Contribution

It constructs specific metrics on countable groups to show infinite Assouad-Nagata dimension when the center is not locally finite, solving existing open problems.

Findings

Groups with non-locally finite centers have infinite Assouad-Nagata dimension under certain metrics.

Provides explicit construction of proper, left-invariant metrics for these groups.

Addresses and resolves two open problems posed by A. Dranishnikov.

Abstract

Suppose $G$ is a countable, not necessarily finitely generated, group. We show $G$ admits a proper, left-invariant metric $d_G$ such that the Assouad-Nagata dimension of $(G,d_G)$ is infinite, provided the center of $G$ is not locally finite. As a corollary we solve two problems of A.Dranishnikov.