Power maps and completions of free groups and of the modular group (Q1357562)

Let \(G\) be a group and \(P_n\colon G\to G\), \(x\mapsto x^n\), \(n\in\mathbb{N}\), be the \(n\)-th power map. If \(K\) is a normal subgroup of \(G\) then let \(P_n(G\bmod K)\) be the preimage of \(P_n(G/K)\) in \(G\). If \({\mathcal K}\) is a family of normal subgroups of \(G\) then let be \(P_n(G\bmod{\mathcal K})=\bigcap_{K\in{\mathcal K}}P_n(G\bmod K)\). For \(G\) let \(\text{Fin }G\) be the set of all normal subgroups of \(G\) of finite index. If \(G=F\) is free then \(P_n(F\bmod\text{Fin }F)=P_n(F)\). Now let \(G=\Gamma=\text{SL}(2,\mathbb{Z})\) be the classical modular group. For each \(N\in\mathbb{N}\) let \(\Gamma_N\) be the principal congruence subgroup of level \(N\). If \({\mathcal K}=\{\Gamma_N\mid N\in\mathbb{N}\}\) then \(P_n(\Gamma\bmod{\mathcal K})\subset P_n(\Gamma)\) for \(n>2\) and \(P_n(\Gamma\bmod\text{Fin }\Gamma)=P_n(\Gamma)\) for all \(n\in\mathbb{N}\).