Let $K$ be the Klein bottle. Give examples of cover spaces $p_1:E_1\to K, p_1:E_2\to K$ each with two sheets and such that $E_1$ and $E_2$ are arc-connected but not isomorphic as covering spaces.
I know that $\pi_1(K)\cong \left \langle a,b:abab^{-1}=1 \right \rangle\cong\mathbb{Z}\triangleleft\mathbb{Z}$, where $\mathbb{Z}\triangleleft\mathbb{Z}$ is the set of pairs $(m,n)$ with $m,n\in \mathbb{Z}$. The product in $\mathbb{Z}\triangleleft\mathbb{Z}$ is defined by $(n_1,n_2)(n_1+(-1)^{m_1}n_2,m_1+m_2)$. The group $G=\mathbb{Z}\triangleleft\mathbb{Z}$ is properly discontinuous on $\mathbb{R}^2$ and thus $\pi:\mathbb{R}^2\to\mathbb{R}^2/G$ is a universal covering. In this case $\mathbb{R}^2/G\cong [0,1]\times[0,1]/\sim$, that is $\mathbb{R}^2/G=K$.
If we take $A=\{(n,2m):m.n\in \mathbb{Z}\}$, how do I show that $A$ is a subgroup of G which has index $2$ in $G$? Also note that the covering $r:\mathbb{R}^2/A\to \mathbb{R}^2/G$ corresponding to the subgroup $A$ is atorus, because $\mathbb{R}^2/A\cong \mathbb{S}^1\times\mathbb{S}^1$.
I can do the same with $B=\{(2n,m):m,n\in\mathbb{Z}\}$ showing that this subgroup has index $2$ in $G$ and that $\mathbb{R}^2/B$ corresponds to Klein's bottle. The thing is that I would like to try all the details, I think I have the idea but I would like to do everything in the best way possible and very rigorously, for example:
Why does this product define a group in $\mathbb{Z}\triangleleft\mathbb{Z}$?
Why $\left \langle a,b:abab^{-1}=1 \right \rangle\cong\mathbb{Z}\triangleleft\mathbb{Z}$?
Why does $G$ continuously act on $\mathbb{R}^2$?
Why is this action properly discontinuous?
Why $A$ and $B$ are subgroups of $G$ with index $2$?
I know that $\mathbb{R}^2/A\cong \mathbb{S}^1\times\mathbb{S}^1$, but why $\mathbb{R}^2/B$ corresponds to Klein's bottle?
Thank you very much.