Henry Bradford and I found some lower bound on the word growth for some groups. We are looking for examples of groups for which this will be useful. So we need examples of finitely generated group which have infinitely many subgroups of finite index, but for each prime $ p$ only finitely many of $ p$ -power index. We would like all of their finitely generated subgroups to have the same property and we would like to have examples where not much is known about their word growth.
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