Embedding relatively hyperbolic groups in products of trees

Mackay, John M; Sisto, Alessandro

doi:10.2140/agt.2013.13.2261

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Embedding relatively hyperbolic groups in products of trees

John M Mackay and Alessandro Sisto

Algebraic & Geometric Topology 13 (2013) 2261–2282

DOI: 10.2140/agt.2013.13.2261

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.

Keywords

relatively hyperbolic group, asymptotic Assouad–Nagata dimension, linearly controlled asymptotic dimension, product of trees

Mathematical Subject Classification 2010

Primary: 20F65, 20F69

References

Publication

Received: 24 November 2012

Revised: 16 January 2013

Accepted: 19 March 2013

Published: 19 June 2013

Authors

John M Mackay
Mathematical Institute University of Oxford 24–29 St Giles’ Oxford OX1 3LB UK

Alessandro Sisto
Mathematical Institute University of Oxford 24–29 St Giles’ Oxford OX1 3LB UK
http://people.maths.ox.ac.uk/sisto/

Volume 13, issue 4 (2013)