Quasi-isometric embedding of the fundamental group of an orthogonal graph-manifold into a product of metric trees

TL;DR

The paper introduces a class of orthogonal graph-manifolds in dimensions three and higher, proving their fundamental groups can be quasi-isometrically embedded into a product of trees, revealing their asymptotic dimensions.

Contribution

It establishes the quasi-isometric embedding of fundamental groups of orthogonal graph-manifolds into products of trees, a novel geometric insight for these manifolds.

Findings

Fundamental groups of orthogonal graph-manifolds embed into products of trees.

Asymptotic and linearly-controlled asymptotic dimensions equal the manifold dimension.

Embedding results hold for all dimensions n ≥ 3.

Abstract

In every dimension $n\ge 3$ we introduce a class of orthogonal graph-manifolds and prove that the fundamental group of any orthogonal graph-manifold quasi-isometrically embeds into a product of $n$ trees. As a consequence, we obtain that asymptotic and linearly-controlled asymptotic dimensions of such group are equal to $n$.