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Distortion of imbeddings of groups of intermediate growth into metric spaces


Authors: Laurent Bartholdi and Anna Erschler
Journal: Proc. Amer. Math. Soc. 145 (2017), 1943-1952
MSC (2010): Primary 20F65; Secondary 51F99
DOI: https://doi.org/10.1090/proc/13453
Published electronically: December 15, 2016
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Abstract: We show that groups of subexponential growth can have arbitrarily bad distortion for their imbeddings into Hilbert space.

More generally, consider a metric space $ \mathcal X$, and assume that it admits a sequence of finite groups of bounded-size generating set which does not imbed coarsely in $ \mathcal X$. Then, for every unbounded increasing function $ \rho $, we produce a group of subexponential word growth all of whose imbeddings in $ \mathcal X$ have distortion worse than $ \rho $.

This implies that Liouville groups may have arbitrarily bad distortion for their imbeddings into Hilbert space, precluding a converse to the result by Naor and Peres that groups with distortion much better than $ \sqrt t$ are Liouville.


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Additional Information

Laurent Bartholdi
Affiliation: Département de mathématiques et applications, École Normale Supérieure, Paris, France – and – Mathematisches Institut, Georg-August Universität, Göttingen, Germany
Email: laurent.bartholdi@gmail.com

Anna Erschler
Affiliation: C.N.R.S., Département de mathématiques et applications, École Normale Supérieure, Paris, France
Email: anna.erschler@ens.fr

DOI: https://doi.org/10.1090/proc/13453
Received by editor(s): March 19, 2015
Received by editor(s) in revised form: July 4, 2016
Published electronically: December 15, 2016
Additional Notes: This work was supported by the ERC starting grant 257110 “RaWG”, the ANR “DiscGroup: facettes des groupes discrets”, the ANR “@raction” grant ANR-14-ACHN-0018-01, the Centre International de Mathématiques et Informatique, Toulouse, and the Institut Henri Poincaré, Paris
Communicated by: Kevin Whyte
Article copyright: © Copyright 2016 by the authors

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