Linearly-controlled asymptotic dimension of the fundamental group of a graph-manifold

TL;DR

This paper establishes an upper bound of 7 on the linearly controlled asymptotic dimension of fundamental groups of 3D graph-manifolds, with implications for their universal covers' geometric properties.

Contribution

It provides the first explicit bound on the asymptotic dimension of these groups and demonstrates their universal covers' Lipschitz retract and quasisymmetric embedding.

Findings

Asymptotic dimension of fundamental groups is at most 7.

Universal covers are absolute Lipschitz retracts.

Universal covers admit quasisymmetric embeddings into product of 8 metric trees.

Abstract

We prove that the linearly controlled asymptotic dimension of the fundamental group of any 3-dimensional graph-manifold does not exceed 7. As applications we obtain that the universal cover of such a graph-manifold is an absolute Lipschitz retract and it admits a quasisymmetric embedding into the product of 8 metric trees.