# Acyclic group

2020 Mathematics Subject Classification: Primary: 20J05 [MSN][ZBL]

A group having the same constant coefficient homology as the trivial group (cf. also Homology). This means that its classifying space is an acyclic space. In the literature the earliest examples are Higman's four-generator four-relator group [Hi]

$$\langle x_0, x_1, x_2, x_3 : x_{i+1}x_ix_{i+1}^{-1} = x_i^2, i\in \mathbb{Z}/4\rangle$$

and others found in combinatorial group theory [BaGr], [BaDyHe], [BeMi]. Further examples arise in geometry ([Ep], [Ma], [Se], [SaVa], [GrSe]) or as automorphism groups of large objects ([HaMc]; for example, the group of all bijections of an infinite set). Algebraically closed groups are acyclic.

Many proofs of acyclicity of infinitely generated groups rely on the property that all binate groups are acyclic [Be3] (cf. also Binate group). An important result in the plus-construction approach to the higher algebraic $K$-theory of rings and operator algebras is that the infinite general linear group of the cone of a ring is acyclic [Wa], [Be]. Topologically, the plus-construction of a topological space is completely determined by a certain perfect, locally free, and hence acyclic, group [BeCa].

Ubiquity results for acyclic groups include the following:

• Every group is a normal subgroup of a normal subgroup of an acyclic group. This result has applications to algebraic topology [KaTh].

In contrast to the above are results indicating that acyclic groups have "few" normal subgroups. Thus, the following acyclic groups admit no non-trivial finite-dimensional linear representations over any field:

• algebraically closed groups;
• Higman's group [Hi];
• torsion-generated acyclic groups [Be4];

Moreover, many of the above groups are simple modulo the centre.

How to Cite This Entry:
Acyclic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Acyclic_group&oldid=52059
This article was adapted from an original article by A.J. Berrick (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article