On the Maximal Size of Irredundant Generating Sets in Lie Groups and Algebraic Groups

We show the following dichotomy for a connected Lie group $G$: If $G$ is amenable, then any topologically generating set $X\subset G$ of size larger than a fixed polynomial in the dimension of $G$ must be redundant (i.e., a proper subset of $X$ still generates $G$). If $G$ is non-amenable, then it admits arbitrarily large topologically generating sets that are irredundant, and remain irredundant even after applying Nielsen transformations.
The polynomial bound for amenable groups is obtained by reduction to finite simple groups of Lie type via strong approximation. This partially answers two conjectures by Gelander on generation in compact Lie groups and simple algebraic groups, and moreover shows that these conjectures are implied by the Wiegold conjecture.
The construction of large Nielsen irredundant generating sets in non-amenable groups is done by extending Minsky's work to higher rank Lie groups, exhibiting dense representations in the domain of discontinuity of the $\mathrm{Out}(F_{n})$-action on the character variety.