Four-dimensional graph-manifolds with fundamental groups quasi-isometric to fundamental groups of orthogonal graph-manifolds

TL;DR

This paper introduces a topological invariant for 4-dimensional graph-manifolds and proves that those with a certain type are bi-Lipschitz equivalent to orthogonal graph-manifolds in their universal covers.

Contribution

It defines a new topological invariant for 4D graph-manifolds and establishes a bi-Lipschitz equivalence result for manifolds of type at most two.

Findings

Universal covers of certain 4D graph-manifolds are bi-Lipschitz equivalent to orthogonal graph-manifolds.

The introduced invariant classifies 4D graph-manifolds into types.

Manifolds of type ≤ 2 share geometric properties with orthogonal graph-manifolds.

Abstract

We introduce a topological invariant, it a type of a graph-manifold, which takes natural values. For a 4-dimensional graph-manifold, whose type does not exceed two, it is proved that its universal cover is bi-Lipschitz equivalent to a universal cover of an orthogonal graph-manifold (for any Riemannian metrics on graph-manifolds).