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  • From Symmetries of the Modular Tower of Genus Zero Real Stable Curves to a Euler Class for the Dyadic Circle

  • Christophe Kapoudjian
  • Journal:
    Compositio Mathematica / Volume 137 / Issue 1 / May 2003
    Published online by Cambridge University Press:
    04 December 2007, pp. 49-73
    Print publication:
    May 2003
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  • We construct actions of the spheromorphism group of Neretin (containing Thompson's group V) 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 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’, can be viewed as analogues of the Artin pure braid groups. By lifting the actions of the Thompson and Neretin groups to the universal covers of the towers, we obtain extensions of both groups by an infinite pure quasi-braid group, and construct an ‘Euler class’ for the Neretin group. We justify this terminology by constructing a corresponding cocycle.