Volume 13, issue 4 (2013)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Embedding relatively hyperbolic groups in products of trees

John M Mackay and Alessandro Sisto

Algebraic & Geometric Topology 13 (2013) 2261–2282
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/