# A note on LERF groups and generic group actions Academic Article

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### 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.

### publication date

• January 1, 2014