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 : PA → GA such that i induces an isomorphism of profinite completions, where GA is a torsion-free biautomatic group that is residually finite and superperfect, while PA is a finitely generated group with H2(PA,ℤ) ≅ A.