Use the <Schreier coset graph> of the subgroup, with one directed edge labelled $s$ from $Hg$ to $Hgs$ for each $s\in\{a,b\}$. The homomorphism
$$
\phi:F_2\longrightarrow\mathbb Z,
\qquad
\phi(a)=1,\quad\phi(b)=0
$$
identifies the cosets of $\ker\phi$ with the integers. The covering graph therefore has vertices $v_n$, $n\in\mathbb Z$, an $a$-edge
$$
v_n\xrightarrow{\ a\ }v_{n+1}
$$
and a $b$-loop at every $v_n$. Thus it is a doubly infinite $a$-line with one $b$-circle attached at each integer vertex.

Solved by gpt-5.6-sol high.


 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-133/1/a/i/solution