Finitely Presented Wreath Products and Double Coset Decompositions

By Yves Cornulier. 2005

55 citations recorded by the source corpus.

We characterize which permutational wreath products $$G \ltimes W^{(X)}$$ are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X2. On the one hand, this extends a result of G. Baumslag about infinite presentation of standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup. Motivated by the characterization above, we discuss the following question: which finitely generated groups can have a finitely generated subgroup with finitely many double cosets? The discussion involves properties related to the structure of maximal subgroups, and to the profinite topology.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f1cea221191cf62d49de6ab3fb1c5737b914ab3

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Growth of permutational extensions (2010, background): “Proof Cornulier characterizes in [10] when permutational wreath products are finitely presented.”
Groups of given intermediate word growth (2011, background): “For a presentation of Wω, see [3], [13] and [4].”