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Instability of nondiscrete free subgroups in lie groups

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Abstract

We study finitely-generated nondiscrete free subgroups in Lie groups. We address the following question first raised by Étienne Ghys: is it always possible to make arbitrarily small perturbations of the generators of the free subgroup in such a way that the new group formed by the perturbed generators be not free? In other words, is it possible to approximate generators of a free subgroup by elements satisfying a nontrivial relation? We prove that the answer to Ghys’ question is positive and generalize this result to certain nonfree subgroups. We also consider the question on the best approximation rate in terms of the minimal length of relation in the approximating group. We give an upper bound on the optimal approximation rate as \( {e^{ - c{l^\kappa }}} \), where c > 0 is a constant, l the minimal length of relation and 0.19 < κ < 0.2.

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Authors and Affiliations

  1. Laboratoire J.-V.Poncelet, UMI 2615 du CNRS et l’Université Indépendante de Moscou, Moscow, Russia

    Alexey Glutsyuk

  2. CNRS, Unité de Mathématiques Pures et Appliquées, M.R. École Normale Supérieure de Lyon, 46 allée d’Italie, 69364, Lyon 07, France

    Alexey Glutsyuk

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  1. Alexey Glutsyuk
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Correspondence to Alexey Glutsyuk.

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Supported in part by RFBR grants 02-01-00482, 02-01-22002, 07-01-00017-a and NTsNIL_a (RFBR-CNRS) 05-01-02801, NTsNIL_a (RFBR-CNRS) 10-01-93115.

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Glutsyuk, A. Instability of nondiscrete free subgroups in lie groups. Transformation Groups 16, 413–479 (2011). https://doi.org/10.1007/s00031-011-9134-9

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