On the girth of groups acting on CAT(0) cube complexes

Le résumé fourni par la source

We obtain a sufficient condition for lattices in the automorphism group of a finite-dimensional CAT(0) cube complex to have infinite girth.As a corollary, we get a version of the girth alternative for groups acting geometrically on CAT(0) cube complexes: any such group is either {locally finite}-by-{virtually abelian} or it has infinite girth.We produce counterexamples to show that the alternative fails in the general class of groups acting cocompactly on finite-dimensional CAT(0) cube complexes using examples of nonvirtually solvable groups which satisfy a law.

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