Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 21F / c / Solution

Let $F_2=⟨a,b⟩={\pi }_1(S^1\vee S^1)$ and take the connected two-sheeted covering graph corresponding to the kernel ofIt has two vertices, a $b$-loop at each vertex, and two lifted $a$-edges joining the vertices. Choose one lifted $a$-edge as a spanning tree. The other three edges give the based loopsCollapsing the spanning tree identifies the cover with a wedge of three circles, so these loops freely generate its fundamental group. Hence they generate an explicit subgroupisomorphic to $F_3$, in agreement with the Reidemeister–Schreier theorem.