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

Published online by Cambridge University Press:  04 December 2007

Christophe Kapoudjian
Affiliation:
Laboratoire Emile Picard, UMR 5580, University of Toulouse III, 31062 Toulouse cedex 4, France. e-mail: ckapoudj@picard.ups-tlse.fr
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Abstract

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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.

Type
Research Article
Copyright
© 2003 Kluwer Academic Publishers