The classical Hurewicz theorem for topological dimension states that if \(f\colon X\to Y\) is a continuous map where \(X\) and \(Y\) are compact metric spaces and if for every \(y\in Y\) the topological dimension of the preimage \(f^{-1}(y)\) does not exceed some fixed integer \(n\), then \(\dim X\leq\dim Y+n\).
In this paper, the asymptotic analogue of the theorem is proven: Let \(f\colon X\to Y\) be a Lipschitz map of a geodesic metric space \(X\) to a metric space \(Y\). Suppose that for every \(R>0\) the asymptotic dimension of preimages \(f^{-1}(B_r(y))\) of \(R\)-balls in \(Y\) satisfies the inequality \(\text{asdim}\leq n\) uniformly. Then \(\text{asdim\,}X\leq\text{asdim\,}Y+n\). – For the proof of the theorem the authors construct for arbitrary \(\varepsilon>0\) a uniformly cobounded \(\varepsilon\)-Lipschitz map \(\phi\colon X\to Y\) to a uniform simplicial complex of dimension \(\leq\text{asdim\,}X+n\).
The theorem has a number of applications to geometric group theory. In particular, it allows estimating the asymptotic dimension of a finitely generated group acting by isometries on a metric space, of amalgamated products of groups, etc. Also, there are estimations \(\text{asdim\,}G\leq h(G)\) for the asymptotic dimension of a finitely generated polycyclic or nilpotent group or nilpotent Lie group \(G\) in terms of its Hirsch length \(h(G)\).