Universal groups, binate groups and acyclicity

Remarks on the Structure of Acyclic Groups

“…(1.1) [8] Any group G may be normally embedded in a group which is itself a normal subgroup of an acyclic group.…”

“…In recent years acyclic groups have received increasing attention as a result of their importance in algebraic K-theory, foliation theory, cohomology of groups and elsewhere. See [8,10] for collections of examples.…”

The acyclic group dichotomy

“…We begin with an elementary observation, that follows readily from the formula A nontrivial group G is called binate [8] if for any finitely generated subgroup H of G there exists a homomorphism (called a structure map) ϕ = ϕ H : H → G and element (called a structure element ) u = u H ∈ G such that for all h ∈ H,…”

“…We turn now to the right-hand half of the flow diagram (1-1). A universal method for embedding any group H in a binate group is the universal binate tower on H, constructed in [8] by means of HNN-extensions, as follows. Let H 0 = H and for each i ≥ 0…”

From acyclic groups to the Bass conjecture for amenable groups

“…In the extreme case where G is itself abelian, a weaker form of this was already known to be possible by making G the centre of an acyclic group [4], [7], [8]. A prominent class of acyclic groups suited to our purpose comprises the binate groups [8]. We now recall the definition.…”