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We say that a subgroup of ${\rm Sym}(\mathbb{N})$ has sparse orbit representatives if it has infinitely many orbits on $\mathbb{N}$, but the set of smallest orbit representatives has natural density 0 (and in particular its natural density exists).

Which are the with respect to inclusion largest subgroups of ${\rm Sym}(\mathbb{N})$ which do not have finitely generated subgroups with sparse orbit representatives?

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  • $\begingroup$ What's the relation with the Collatz conjecture ? $\endgroup$
    – Sebastien Palcoux
    Mar 4, 2014 at 22:34
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    $\begingroup$ @SébastienPalcoux: Let $G$ be the group mentioned under "Ad 2" in mathoverflow.net/questions/112527, and let $H$ be the permutation group induced by $G$ on $\mathbb{N}$. If $H$ is a subgroup of one of the groups this question asks for, then the Collatz graph has only finitely many connected components. -- That said, the question is much more general, and a sufficiently explicit answer could be of interest well beyond the Collatz conjecture. $\endgroup$
    – Stefan Kohl ♦
    Mar 4, 2014 at 23:48

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