 Solution

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 homomorphismidentifies the cosets of $\ker \varphi$ with the integers. The covering graph therefore has vertices $v_n$, $n\in \mathbb{Z}$, an $a$-edgeand a $b$-loop at every $v_n$. Thus it is a doubly infinite $a$-line with one $b$-circle attached at each integer vertex.