  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 $(\tau)$. 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.
