Question

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.

Answer 1

These groups are virtually wreath products of finitely generated Abelian groups with $\mathbb Z$. More precisely the subgroup $G_s=\langle a^s, b\rangle$ is a wreath product of a 1-related Abelian group (the relation is your product of powers) and $\langle a^s\rangle$, and your group is an extension of $G_s$ by a cyclic group of order $s$. I do not think there exists a special name for these groups. As with every metabelian group which is Abelian-by-cyclic, you can get a lot of properties of it by looking at the corresponding $\mathbb{Z}[b,b^{-1}]$-module.

Update. I just realized that this group is not a virtual wreath product. One can construct the group as follows. Consider the ring $R=\mathbb{Z}[t,t^{-1}]$ and the ideal $I$. there generated by the polynomial $f=c_0+c_1t+\ldots+c_st^s$. The group $\mathbb{Z}$ acts on the Abelian group $R/I$ by left multiplication (which are automorphisms of the additive group of $R/I$). Your group is the corresponding semidirect product $R/I\rtimes \mathbb{Z}$. One can also represent this group by $2\times 2$ matrices using the Magnus embedding. Anyway this does not answer your question about the name of the group. I still do not think it has a name.