Abstract: We consider contact processes on the hierarchical group $\om_N$ with freedom $N$, where sites infect other sites at hierarchical distance $k$ with rate $\al_kN^{-k}$, and sites become healthy with recovery rate $\de$. We show that the critical recovery rate $\de_{\rm c}$ is zero (i.e., the process dies out for any $\de>0$) if $\liminf_{k\to\infty}N^{-k}\log(\bet_k)=-\infty$, where $\bet_k:=\sum_{n=k}^\infty\al_n$. On the other hand, in the special case that $N$ is a power of two, we show that $\de_{\rm c}>0$ provided that $\sum_kN^{-k}\log(\al_k)>-\infty$. The proof of this latter fact is based on a coupling argument that compares contact processes on $\om_2$ with contact processes on a renormalized lattice.