Subgroups of GL(2,C) isomorphic to Z Let $\mathbb Z\to \mathrm{GL}_2(\mathbb C)$ be an injective homomorphism. I'm wondering about the possibilities for the image of $\mathbb Z$.
I think the image is always conjugate to a subgroup of matrices of the form $$\left( \begin{array}{cc} \lambda_1 & b \\ 0 & \lambda_2\end{array}\right),$$ where $b \in n\mathbb Z$ (for some $n$) and $\lambda_1$ and $\lambda_2$ are in $\mathbb C$?
My question is what other non-trivial and easy to state conditions does this cyclic subgroup (of infinite order) have to satisfy?
If $b=0$, then $\lambda_1$ or $\lambda_2$ is of infinite order (in $\mathbb C^\ast$) and both are non-zero.
What if $b\neq 0 $?

The Jordan normal form of the image of $1\in\mathbb Z$ is either
$$\begin{pmatrix}\lambda_1&0\\0&\lambda_2\end{pmatrix}, $$
 which is the case you handled, or it is 
$$\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}. $$
In the second case, 
$$\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}^n=\begin{pmatrix}\lambda^n&n\lambda^{n-1}\\0&\lambda^n\end{pmatrix}, $$
so any (nonzero) $\lambda$  will do.
