  Bartholdi, Neuhauser and Woess proved that a family of metabelian groups
including lamplighters have a striking geometric manifestation as 1-skeleta of
horocyclic products of trees. The purpose of this article is to give an
elementary account of this result, to widen the family addressed to include the
infinite valence case (for instance $\mathbb{Z} \wr \mathbb{Z}$), and to make
the translation between the algebraic and geometric descriptions explicit.
  In the rank-2 case, where the groups concerned include a celebrated example
of Baumslag and Remeslennikov, we give the translation by means of a
combinatorial `lamplighter description'. This elucidates our proof in the
general case which proceeds by manipulating polynomials.
  Additionally, we show that the Cayley 2-complex of a suitable presentation of
Baumslag and Remeslennikov's example is a horocyclic product of three trees.
