Finitely presented wreath products and double coset decompositions

We characterize which permutational wreath products W^(X)\rtimes G 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 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.Comment: 21 pages; no figure. To appear in Geom. Dedicat

The homology of groups, profinite completions, and echoes of Gilbert Baumslag

We present novel constructions concerning the homology of finitely generated groups. Each construction draws on ideas of Gilbert Baumslag. There is a finitely presented acyclic group $U$ such that $U$ has no proper subgroups of finite index and every finitely presented group can be embedded in $U$. There is no algorithm that can determine whether or not a finitely presentable subgroup of a residually finite, biautomatic group is perfect. For every recursively presented abelian group $A$ there exists a pair of groups $i:P_A\hookrightarrow G_A$ such that $i$ induces an isomorphism of profinite completions, where $G_A$ is a torsion-free biautomatic group that is residually finite and superperfect, while $P_A$ is a finitely generated group with $H_2(P_A,\mathbb{Z})\cong A$.Comment: Final version, to appear in volume in honour of Baumslag: "Elementary Theory of Group Rings, and Related Topics