Embedding universal covers of graph manifolds in products of trees

TL;DR

This paper proves that universal covers of graph manifolds can be embedded into a product of three trees, confirming a conjecture about their geometric dimension and providing new insights into their structure.

Contribution

It establishes a quasi-isometric embedding of universal covers of graph manifolds into three trees, confirming the Assouad-Nagata dimension conjecture.

Findings

Universal covers of graph manifolds embed into a product of three trees

Confirmed the Assouad-Nagata dimension of these covers is 3

Provided a new geometric perspective on graph manifolds

Abstract

We prove that the universal cover of any graph manifold quasi-isometrically embeds into a product of three trees. In particular we show that the Assouad-Nagata dimension of the universal cover of any closed graph manifold is 3, proving a conjecture of Smirnov.