The acyclic group dichotomy

TL;DR

This paper explores the properties of acyclic groups, especially binate groups, and their relationships with other groups, highlighting their significance in K-theoretic conjectures and group homomorphisms.

Contribution

It establishes the existence of homomorphisms and embeddings involving acyclic and binate groups, revealing their structural roles and limitations.

Findings

Homomorphism from acyclic groups of cohomological dimension 2 onto perfect subgroups

Embedding of any group into a binate (acyclic) group

No nontrivial homomorphisms from binate groups to finite cohomological dimension groups

Abstract

Two extremal classes of acyclic groups are discussed. For an arbitrary group G, there is always a homomorphism from an acyclic group of cohomological dimension 2 onto the maximum perfect subgroup of G, and there is always an embedding of G in a binate (hence acyclic) group. In the other direction, there are no nontrivial homomorphisms from binate groups to groups of finite cohomological dimension. Binate groups are shown to be of significance in relation to a number of important K-theoretic isomorphism conjectures.