Amenable connected Lie groups have polynomial-bounded irredundant topological generating sets; non-amenable ones admit arbitrarily large Nielsen-irredundant ones.
On the maximal size of independent generating sets of PSL2 (q)
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On the Maximal Size of Irredundant Generating Sets in Lie Groups and Algebraic Groups
Amenable connected Lie groups have polynomial-bounded irredundant topological generating sets; non-amenable ones admit arbitrarily large Nielsen-irredundant ones.