A subset S of a group G invariably generates G if G is generated by {sg(s)|s ∈ S} for any choice of g(s) ∈ G,s ∈ S. A topological group G is said to be ℐ𝒢 if it is invariably generated by some subset S ⊆ G, and 𝒯ℐ𝒢 if it is topologically invariably generated by some subset S ⊆ G. In this paper, we study the problem of (topological) invariable generation for linear groups and for automorphism groups of trees. Our main results show that the Lie group SL2(ℝ) and the automorphism group of a regular tree are 𝒯ℐ𝒢, and that the groups PSLm(K),m ≥ 2 are not ℐ𝒢 for countable fields of infinite transcendence degree over a prime field.

中文翻译：

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关于无限群中不变生成的几点说明

一个子集小号一组的G总是产生G如果G由{sG(s)|s ∈ 小号}对于任何选择G(s) ∈ G,s ∈ 小号. 拓扑群G据说是ℐ𝒢如果它总是由某个子集生成小号 ⊆ G， 和𝒯ℐ𝒢如果它在拓扑上总是由某个子集生成小号 ⊆ G. 在本文中，我们研究了线性群和树的自同构群的（拓扑）不变生成问题。我们的主要结果表明，李群SL2(ℝ)并且正则树的自同构群是𝒯ℐ𝒢，并且这些组磷小号大号米(ķ),米 ≥ 2不是ℐ𝒢对于素数域上无限超越度的可数域。