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

The relation isso $K$ is the Klein bottle group. Let it act on ${\mathbb{R}}^2$ byThese are Euclidean isometries and satisfy $ab{a}^{-1}=b^{-1}$. Every element has a normal form $b^m{a}^n$. The orbit of $(0,0)$ is discrete, and a rectangle of finite size meets every orbit, so the action is proper and cocompact.

The squareis a vertical translation, while $b$ is a horizontal translation. They commute, andas an isometry only when $m=n=0$. Hence $⟨a^2,b⟩\cong {\mathbb{Z}}^2$. The normal form shows that every element lies in either $⟨a^2,b⟩$ or $a⟨a^2,b⟩$, so this subgroup has index two in $K$.