Finite dimensional amenable groups

Abstract

We show that an amenable group of finite Assouad-Nagata dimension satisfies the property $H_{FD}$ of Shalom. Such infinite groups are known to admit a virtual homomorphism onto $\mathbb{Z}$, and thus our result implies that an amenable group of finite $AN$-dimension cannot be a simple group. We can also conclude that an amenable group of finite $AN$-dimension cannot be a torsion group. Our proof is based on new estimates of diameters of Følner couples. We prove that any amenable group of finite $AN$-dimension admits Følner couples inside balls of linear diameter and more generally estimate the radius of the balls containing Følner couples in groups of finite asymptotic dimension. This result strengthens the result of Nowak about diameters of Følner sets.