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

## TL;DR

This paper introduces new constructions in the homology of finitely generated groups, including a universal acyclic group and results on the complexity of subgroup properties, drawing on ideas of Gilbert Baumslag.

## Contribution

It presents novel group constructions and demonstrates the complexity of subgroup decision problems, extending Baumslag's ideas in homology and group embeddings.

## Key findings

- Existence of a finitely presented acyclic group with unique subgroup properties
- Undecidability of subgroup perfection in certain residually finite groups
- Construction of groups with prescribed second homology related to any abelian group

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