From wall spaces to CAT(0) cube complexes

Abstract

We explain how to adapt a construction of M. Sageev's to construct a proper action on a CAT(0) cube complex starting from a proper action on a wall space, and use this to deduce that if G is a group containing an amenable subgroup H of super-polynomial growth and G acts properly on a space with walls then there are arbitrarily large families of walls in the wall space which pairwise cross.