= Infinite residually finite torsion group
Enumerate all words $w_i$ on two generators in a <free pro-p group>. Choose powers $p^{e_i}$ so large that $\sum_i\tau^{p^{e_i}}<2\tau-1$ for some $1/2<\tau<1$, and impose the closed normal relations $w_i^{p^{e_i}}=1$. The <Golod-Shafarevich infinitude criterion for pro-p groups> makes the resulting pro-$p$ group infinite. Its abstract subgroup generated by the two generator images is dense and infinite; each element is represented by an enumerated word and has finite order. The ambient finite quotients separate its elements, so it is also a <residually finite group>. The abstract subgroup, rather than the whole topological group, is the desired finitely generated <torsion group>.
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