2014/09/16 by Yair Glasner, Glasner, Yair, Daniel Kitroser +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Geometric and Algebraic Topology #math.GR #msc:0B07 #msc:20E26
paper · pdf · doi:10.48550/arxiv.1409.4737
6 pages
arxiv created 2014/09/16 · arxiv updated 2014/09/17
Let G be a finitely generated group, 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 Sub(G), (iii) A Baire generic homomorphism ϕ: G → X! admits only finite orbits. Property (i) is known as the LERF property. We introduce a new family of groups which we call \itA-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.