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A result of Schreier and Ulam from their 1935 paper "Sur le nombre des g$\acute{\textrm{e}}$n$\acute{\textrm{e}}$rateurs d'un groupe topologique compact et connexe" says that if $G$ is a connected compact second countable group then the set $D=\{ (g,h)\in G^2 : \overline{\langle g,h\rangle} = G\}$ of pairs generating a dense subgroup is comeager in $G^2$ (in fact, $D$ is dense $G_\delta$, the $G_\delta$ part is clear though). Here $\langle g,h\rangle$ denotes the subgroup generated by $g$ and $h$.

Unfortunately, at least one language barrier is making it difficult for me to understand the proof given in this paper. (While I am somewhat competent in French, S&U in their proof refer to results and notation in a paper by von Neumann that is written in German.)

I am looking for another exposition of this result. So far my own web searches have been unsuccessful.

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up vote 1 down vote accepted

The paper "Dense embeddings of surface groups" by Emmanuel Breuillard, Tsachik Gelander, Juan Souto, and Peter Storm (Geometry & Topology 10 (2006) 1373–1389) gives a proof in section 8.

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