Embedding relatively hyperbolic groups in products of trees

TL;DR

This paper proves that relatively hyperbolic groups can be embedded into products of finitely many trees if their peripheral subgroups can, and applies this to estimate the asymptotic dimension of 3-manifold groups.

Contribution

It establishes a quasi-isometric embedding of relatively hyperbolic groups into products of trees based on their peripheral subgroups, with applications to 3-manifold groups.

Findings

Relatively hyperbolic groups embed into products of finitely many trees.

Fundamental groups of closed 3-manifolds have asymptotic dimension at most 8.

Fundamental groups of Haken 3-manifolds with boundary have asymptotic dimension 2.

Abstract

We show that a relatively hyperbolic group quasi-isometrically embeds in a product of finitely many trees if the peripheral subgroups do, and we provide an estimate on the minimal number of trees needed. Applying our result to the case of 3-manifolds, we show that fundamental groups of closed 3-manifolds have linearly controlled asymptotic dimension at most 8. To complement this result, we observe that fundamental groups of Haken 3-manifolds with non-empty boundary have asymptotic dimension 2.