We introduce the notion of a telescope of groups. Very roughly a telescope is
a directed system of groups that contains various commuting images of some
fixed group B. Telescopes are inspired from the theory of groups acting on
rooted trees. Imitating known constructions of branch groups, we obtain a
number of examples of B-telescopes and discuss several applications. We give
examples of 2-generated infinite amenable simple groups. We show that every
finitely generated residually finite (amenable) group embeds into a finitely
generated (amenable) LEF simple group. We construct 2-generated frames in
products of finite simple groups and show that there are Grothendieck pairs
consisting of amenable groups and groups with property (Ï„). We give
examples of automorphisms of finitely generated, residually finite, amenable
groups that are not inner, but become inner in the profinite completion. We
describe non-elementary amenable examples of finitely generated, residually
finite groups all of whose finitely generated subnormal subgroups are direct
factors.Comment: 41 pages, comments welcom