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Regular version of the site

Article

Instability of nondiscrete free subgroups in Lie groups

Transformation Groups. 2011. Vol. 16. No. 2. P. 413-479.

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 that is exponential in the  r-th power of the minimal length of relation, where 0.19<r<0.2.