A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory

Bell, G.; Dranishnikov, A.

doi:10.1090/S0002-9947-06-04088-8

A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory
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by G. C. Bell and A. N. Dranishnikov PDF

Trans. Amer. Math. Soc. 358 (2006), 4749-4764 Request permission

Abstract:

We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite-dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.

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