The presentation is a bit prettier if we set $b_i := b^{a^i}$ for $i\in \mathbb{Z}$:

$\langle a, b \mid [b, b_i] \; (i\in \mathbb{Z}), b_0^{c_0}b_1^{c_1}\ldots b_s^{c_s}\rangle$,

Since the subgroup generated by the $b_i$ is abelian, we can even write the last relator additively as $\sum_{i=0}^s c_ib_i$.

These groups occur in a paper Derek Holt and I are writing on subgroups of finitely generated soluble groups. They are all metabelian, and if the $c_i$ are relatively prime, they are also torsion-free.

I would like to call these groups by their proper name if they have one, especially as we have not so far managed to come up with a good name. Since they are fairly straightforward groups, I expect it is possible they have already come up somewhere else and been named, but unfortunately presentations are difficult to search for on MathSciNet.

EDIT: I might as well mention our current name for these groups. We use the notation $G(\mathbb{c})$, where $\mathbb{c} = (c_0,\ldots,c_s)$, for the group with the above presentation. This I am happy with. But our current name for the groups in general is `Gc-groups', which I don't like very much, and so I was hoping to find out that they already have a better name.