  We consider contact processes on the hierarchical group, where sites infect
other sites at a rate depending on their hierarchical distance, and sites
become healthy with a constant recovery rate. If the infection rates decay too
fast as a function of the hierarchical distance, then we show that the critical
recovery rate is zero. On the other hand, we derive sufficient conditions on
the speed of decay of the infection rates for the process to exhibit a
nontrivial phase transition between extinction and survival. For our sufficient
conditions, we use a coupling argument that compares contact processes on the
hierarchical group with freedom two with contact processes on a renormalized
lattice. An interesting novelty in this renormalization argument is the use of
a result due to Rogers and Pitman on Markov functionals.
