Hurewicz Theorem for Nagata-Assouad dimension
by
Nikolay Brodskiy
University of Tennessee
Coauthors: Jerzy Dydak, Michael Levin, Atish Mitra

Given a function f: X→ Y of metric spaces, its asymptotic dimension asdim(f) is the supremum of asdim(A) such that A ⊂ Y and asdim(f(A))=0. Our main result is

Theorem A: asdim(X) ≤ asdim(f)+asdim(Y) for any coarse function f: X→ Y.

As an application we prove

Theorem B: asdim(G) ≤ asdim(K)+asdim(H) for any short exact sequence 1→ K→ G→ H→ 1 of countable groups.

Both Theorems A and B generalize results of Bell and Dranishnikov in which f is Lipschitz and X is geodesic and G, K are finitely generated, respectively. We provide analogs of A and B for linearly controlled asymptotic dimension and Nagata-Assouad dimension.

Date received: February 28, 2006