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.

v1: 18 pages; v2: 20 pages, minor changes

References in corpus (2)

• Embedding mapping class groups into finite products of trees
• Assouad-Nagata dimension of tree-graded spaces

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• Asymptotic Dimension of Graphs of Groups and One Relator Groups
• Property (QT) for 3-manifold groups
• Asymptotic dimension and geometric decompositions in dimensions 3 and 4