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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Distortion of imbeddings of groups of intermediate growth into metric spaces
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by Laurent Bartholdi and Anna Erschler PDF
Proc. Amer. Math. Soc. 145 (2017), 1943-1952

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.

References
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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
  • 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
  • © Copyright 2016 by the authors
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 1943-1952
  • MSC (2010): Primary 20F65; Secondary 51F99
  • DOI: https://doi.org/10.1090/proc/13453
  • MathSciNet review: 3611311