Finitely Presented Wreath Products and Double Coset Decompositions

Abstract

Abstract. We characterize which permutational wreath products G ⋉ 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.

Finiteness properties of totally disconnected locally compact groups

“…Then B A Γ H is the abstract and topological colimit of the sequence (B n A Γ H) of compactly generated groups by Proposition 6.3(i), which does not stabilise, giving a contradiction. Now [15,Lemma 2.9] holds in the context of t.d.l.c. groups, so we may imitate the proof of [15,Proposition 2.10].…”

Groups of given intermediate word growth

“…For a presentation of W ω , see [3], [13] and [4]. Let A be generated by a finite set S; then W ω is generated by S {a, b, c, d}.…”

Growth of permutational extensions

“…Note that ψ 0 , ψ 1 are not homomorphisms, but their restriction to B := b, c, d is a homomorphism; ψ 0 maps to a , while ψ 1 permutes cyclically Proof Cornulier characterizes in [10] when permutational wreath products are finitely presented. A permutational wreath product A X G, for G acting transitively on a transitive G-set X = P \G, is presented as follows: as generators, take those of A and G. As relations, take: those of A and G; the relation [a, u] for every generator a of A and every u in a generating set U of P ; and the relations [a, b t ] for every generators a, b of A and q in a set of double coset representatives of P in G; namely t ∈ T with G = P g∈T P gP .…”