Represent a group in the form of generator and relation

Question

The only nonabelian semidirect product of $\mathbb Z\rtimes_{\theta}\mathbb Z$ is when $\theta(1)=-1$. And write it in formula, we have $$(a,b)(c,d)=(a(-1)^{b}c,bd).$$ But I can't find a good way to represent it as generators and relations. Could anyone give me some help?

Answer 1

The group is isomorphic to the fundamental group $\pi_1(K)$ of the Klein bottle, see also here. By the application of Van Kampen's Theorem to two dimensional CW complexes we obtain the presentation $$ \pi(K) =\left <a,b|abab^{-1}=1\right >. $$