@techreport{b35a316fe23f478995816ff219c26a92,

title = "A note on LERF groups and generic group actions",

abstract = " Let $G$ be a finitely generated group, $\mathrm{Sub}(G)$ the (compact, metric) space of all subgroups of $G$ with the Chaubuty topology and $X!$ the (Polish) group of all permutations of a countable set $X$. We show that the following properties are equivalent: (i) Every finitely generated subgroup is closed in the profinite topology, (ii) the finite index subgroups are dense in $\mathrm{Sub}(G)$, (iii) A Baire generic homomorphism $\phi: G \rightarrow X!$ admits only finite orbits. Property (i) is known as the LERF property. We introduce a new family of groups which we call {\it{A-separable}} groups. These are defined by replacing, in (ii) above, the word {"}finite index{"} by the word {"}co-amenalbe{"}. The class of A-separable groups contains all LERF groups, all amenable groups and more. We investigate some properties of these groups. ",

keywords = "math.GR, Primary 20E26, Secondary 0B07",

author = "Yair Glasner and Daniel Kitroser",

note = "6 pages",

year = "2014",

language = "???core.languages.en_GB???",

series = "Arxiv preprint",

edition = " arXiv:1409.4737 [math.GR]",

type = "WorkingPaper",

}