Instability of nondiscrete free subgroups in lie groups
Instability of nondiscrete free subgroups in lie groups
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李群中非离散自由子群的不稳定性
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发表时间：
2004
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通讯作者：
A. Glutsyuk
中科院分区：
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作者：
A. Glutsyuk
研究李群中有限生成的非离散自由子群。我们解决以下问题首先提出的艾蒂安Ghys：它总是可能使任意小扰动的发电机的自由子群的方式，新的一组所形成的扰动发电机不自由？换句话说，是否有可能通过满足非平凡关系的元素来逼近自由子群的生成元？我们证明了Ghys问题的答案是肯定的，并将这一结果推广到某些非自由子群。我们还考虑了最佳逼近速度问题的最小长度的关系在近似组。我们给出了最优逼近率的上界为$$ {e^{ - c{l^kappa }$$，其中c > 0是常数，l是关系的最小长度，0.19 < κ < 0.2。
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.