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

Abstract

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 $H_2(P_A,\mathbb{Z})\cong A$.

Final version, to appear in volume in honour of Baumslag: "Elementary Theory of Group Rings, and Related Topics"

References in corpus (1)

• On the finite presentation of subdirect products and the nature of residually free groups

Cited by in corpus (1)

• Profinite isomorphisms and fixed-point properties