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Article Dans Une Revue International Journal of Algebra and Computation Année : 2022

Dead ends on wreath products and lamplighter groups

Résumé

For any finite group $A$ and any finitely generated group $B$, we prove that the corresponding lamplighter group $A\wr B$ admits a standard generating set with unbounded depth, and that if $B$ is abelian then the above is true for every standard generating set. This generalizes the case where $B=\mathbb{Z}$ together with its cyclic generator due to Cleary and Taback. When $B=H*K$ is the free product of two finite groups $H$ and $K$, we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of $H$ and $K$. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.

Dates et versions

hal-03903131 , version 1 (16-12-2022)

Identifiants

Citer

Eduardo Silva. Dead ends on wreath products and lamplighter groups. International Journal of Algebra and Computation, 2022, pp.1-42. ⟨10.1142/S0218196723500571⟩. ⟨hal-03903131⟩
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