Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 133 / 1

PutFor every nonzero integer $n$,LetThese are disjoint nonempty subsets of ${\mathbb{R}}^2$. If $(x,y)\in B$, thenso $u^n(B)\subseteq A$. Similarly, if $(x,y)\in A$, thenso $v^n(A)\subseteq B$.

The ping-pong lemma now identifies the subgroup generated by $u$ and $v$ withBoth generators have infinite order, so this is a free group of rank two inside the special linear group ${\mathrm{SL}}_2(\mathbb{Z})$.

Use the injective group homomorphism $F_2\hookrightarrow {\mathrm{SL}}_2(\mathbb{Z})$ from part (a). If $1\ne g\in F_2$, its image is an integral matrix $M\ne I$. Choose a prime number $p$ that does not divide one nonzero entry of $M-I$. The reduction modulo a prime in an integral matrix group homomorphismthen sends $M$ to a nonidentity element. Its target is a finite group, so the composite map separates $g$ from the identity. Hence $F_2$ is a residually finite group.