JournalslemVol. 61, No. 3/4pp. 373–415

Lamplighters, metabelian groups, and horocyclic products

  • Margarita Amchislavska

    Cornell University, Ithaca, USA
  • Timothy R. Riley

    Cornell University, Ithaca, USA
Lamplighters, metabelian groups, and horocyclic products cover
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Abstract

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 ZZ\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.

Cite this article

Margarita Amchislavska, Timothy R. Riley, Lamplighters, metabelian groups, and horocyclic products. Enseign. Math. 61 (2015), no. 3/4, pp. 373–415

DOI 10.4171/LEM/61-3/4-5