Embedding relatively hyperbolic groups in products of trees

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.