What can we say about groups $G$ with $H_3(G)=0$?

Question

Let $G$ be a group. What can we say about groups such that $H_3(G)=0$? If a characterization is not possible, then knowing examples of such groups would be good? Any help is appreciated. Thanks

Answer 1

For example, $H_3(\mathbb{Z})=0$, since for $G=\mathbb{Z}$ the group ring $\mathbb{Z}[G]$ consists of Laurent polynomials $\mathbb{Z}[t,t^{-1}]$, and a free resolution of $\mathbb{Z}$ over $\mathbb{Z}[G]$ is given by $$ 0\rightarrow \mathbb{Z}[t,t^{-1}] \xrightarrow{1-t} \mathbb{Z}[t,t^{-1}] \xrightarrow{t\mapsto 1} \mathbb{Z}. $$ Since it has only length $2$, it follows that $H_n(\mathbb{Z})=H^n(\mathbb{Z})=0$ for all $n\ge 2$.