From symmetries of the modular tower of genus zero real stable curves to an Euler class for the dyadic circle

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.