Asymptotic dimension of 3-dimensional CAT(0) manifolds

Abstract

We prove that every $3$-manifold equipped with a proper metric admitting a convex geodesic bicombing has Assouad--Nagata dimension at most $3$. In particular, every proper CAT$(0)$ $3$-manifold has Assouad--Nagata dimension at most $3$. As a corollary one obtains that the isoperimetric gap theorem of Lang-Stadler-Urech (\cite{LSU}) and Peteranderl\cite{Pet}, generalizes from 3-dimensional Hadamard manifolds to 3-dimensional CAT(0) manifolds.