  We build actions of Thompson group V (related to the Cantor set) and of the
so-called "spheromorphism" group of Neretin, on "towers" of moduli spaces of
genus zero real stable curves. The latter consist of inductive limits of spaces
which are the real parts of the Grothendieck-Knudsen compactification of the
usual moduli spaces of punctured Riemann spheres. By a result of M. Davis, T.
Januszkiewicz and R. Scott, these spaces are aspherical cubical complexes,
whose fundamental groups, the "pure quasi-braid groups", are some analogues of
the classical pure braid groups. By lifting the actions of Thompson and Neretin
groups to the universal covers of the towers, we get new extensions of both
groups by an infinite pure quasi-braid group, and construct what we call an
"Euler class" for Neretin group, justifying the terminology by exhibiting an
Euler-type cocycle. Further, after introducing the infinite (non-pure)
quasi-braid group, we show that both infinite (non-pure and pure) quasi-braid
groups provide new examples of groups whose classifying spaces, after
plus-construction, are loop spaces.
