text: abstract
  the aim of this paper is to make links between the specker compactifications of a locally compact
  group and its convergence actions on compacta. if g is a convergence group acting on a compactum t
  we prove that the closure of g in the attractor sum g⊔t [14] is a quasi-specker compactification of g.
  together with a theorem due to h. abels [2], this implies that for any convergence action such that the
  limit set λ is totally disconnected there exists a surjective g-equivariant continuous map ends g → λ.
  conversely, when the group is compactly generated we show that any specker compactification gives
  rise to a convergence action. given two minimal convergence actions of a compactly generated group
  on totally disconnected compacta m and n, we can then prove the existence of a minimal compactum
  t admitting a convergence action of g and g-equivariant continuous maps t → m and t → n. we
  end the paper giving an interpretation of the set of ends of a compactly generated group as the
  completion by a “visibility” uniformity [14].