Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 134 / 3 / a

A group $G$ is residually finite if, for every $1\ne g\in G$, there are a finite group $Q$ and a homomorphism $\varphi :G\to Q$ such that $\varphi (g)\ne 1$.

Let $F(S)$ be a free group and let $1\ne w\in F(S)$ be a reduced word. In the rose with one oriented loop for each $s\in S$, the word $w$ determines a nonclosed reduced path from a chosen vertex in the universal covering tree. Its finite image path can be completed to a finite covering graph of the rose: for each label, pair the still unmatched incoming and outgoing edge germs, adding finitely many vertices if necessary. The lift of $w$ remains nonclosed in this finite cover.

Let $H\le F(S)$ be the finite-index subgroup represented by this based cover. Then $w\notin H$. The action of $F(S)$ on the finite set of right cosets $H\backslash F(S)$ gives a homomorphism to a finite symmetric group, and $w$ does not fix the coset $H$. Thus $w$ survives in a finite quotient. Since $w$ was arbitrary, every free group is residually finite.