Instability of nondiscrete free subgroups in Lie groups

Abstract

We study finitely-generated nondiscrete free subgroups in Lie groups. We address the following question first raised by \'Etienne Ghys: is it always possible to make arbitrarily small perturbation 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 non-free 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 $e^{-cl^{\kappa}}$, where $c>0$ is a constant, $l$ the minimal length of relation and $0.19<\kappa<0.2$.