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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?

Stefan Kohl
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  • What's the relation with the Collatz conjecture ? – Sebastien Palcoux Mar 04 '14 at 22:34
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    @SébastienPalcoux: Let $G$ be the group mentioned under "Ad 2" in http://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. – Stefan Kohl Mar 04 '14 at 23:48

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