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Finitely presented wreath products and double coset decompositions. (English) Zbl 1137.20019

Summary: 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 \(X^2\). 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.

MSC:

20E22 Extensions, wreath products, and other compositions of groups
20F05 Generators, relations, and presentations of groups
20B22 Multiply transitive infinite groups
20E28 Maximal subgroups
20F65 Geometric group theory
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