Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 133 / 2 / d

Use the amalgam from part (c), with edge groupAn odd power of $a$ belongs to $A\setminus C$, because $C$ is the index-two translation subgroup of the Klein bottle group $A$. Similarly, an odd power of $c$ belongs to $B\setminus C$. Thusis a reduced alternating word whose syllables lie in $A\setminus C$ and $B\setminus C$. The normal form theorem for an amalgamated free product says that every nonempty reduced alternating word is nonidentity. The displayed element is therefore nontrivial for every $n\ge 1$.