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The Dehn quandle, Q, of a surface was defined via the action of Dehn twists about circles on the surface upon other circles. On the torus, 𝕋², we generalize this to show the existence of a quandle Q̂ extending Q and whose elements are measured geodesic foliations. The quandle action in Q̂ is given by applying a shear along such a foliation to another foliation. We extend some results which related Dehn quandle homology to the monodromy of Lefschetz fibrations. We apply certain quandle 2-cycles...
Complexes of groups over ordered simplicial complexes are generalizations to higher dimensions of graphs of groups. We first relate them to complexes of spaces by considering their classifying space . Then we develop their homological algebra aspects. We define the notions of homology and cohomology of a complex of groups with coefficients in a -module and show the existence of free resolutions. We apply those notions to study extensions of complexes of groups with constant or abelian kernel....
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