(May 2016) A group \(G\) is called residually finite if for every \(g\in G\) with \(g\neq 1\), there is a finite group \(H\) and a (surjective) homomorphism \(\varphi\colon G\to H\) with \(\varphi(g)\neq 1\). Let \(G\) be a residually finite group and let \(X\) be the presentation complex for a presentation of \(G\), with vertex \(x_0\). Show that for any loop \(\gamma\colon I\to X\) at \(x_0\) with \(1\neq[\gamma]\in\pi_1(X,x_0)\), there is a finite-sheeted covering space \(p\colon\widetilde X\to X\) and a basepoint \(\widetilde x_0\in p^{-1}(\{x_0\})\) such that \(\gamma\) does not lift to a loop at \(\widetilde x_0\).
Solution
Identify \[ \pi_1(X,x_0)\cong G. \] Let \(g=[\gamma]\ne1\). Since \(G\) is residually finite, there is a finite group \(H\) and a surjective homomorphism \[ \varphi:G\twoheadrightarrow H \] with \(\varphi(g)\ne1\). Put \[ K=\ker\varphi. \] Then \(K\) has finite index \[ [G:K]=|H|. \]
Because a presentation complex is a connected CW complex, the covering-space classification gives a connected based covering \[ p:(\widetilde X,\widetilde x_0)\to(X,x_0) \] whose associated subgroup is \[ p_*\pi_1(\widetilde X,\widetilde x_0)=K. \] The number of sheets is the subgroup index, so this cover is finite-sheeted.
A based loop at \(x_0\) lifts to a loop at \(\widetilde x_0\) exactly when its homotopy class lies in \(K\). But \[ g=[\gamma]\notin K, \] since \(\varphi(g)\ne1\). Hence the lift of \(\gamma\) beginning at \(\widetilde x_0\) does not end at \(\widetilde x_0\). Thus \(\gamma\) does not lift to a loop at the chosen basepoint.